How hot should the oil be when preparing schnitzel?

I generally use about 1cm deep of olive oil when I'm making schnitzel which may or may not be the right oil but it works for me.

I generally find that if the oil is smoking, it's too hot so I tend to get it to a temperature that is very hot, but not smoking.

Once I get it to that temperature, I don't cook too many schnitzel's at once, generally 2 or maybe 3 max to a pan depending on how big your pan is.

The final thing is that you need to make sure you beat the schnitzel out nice and thin before you crumb it so it's only about 1/2 a cm thick when you're cooking it. This way it'll only take a minute or two to cook through and won't overcook the crumb.

A recent episode of America's Test Kitchen recommended heating the oil to 375F and to gently shake the pan back and forth while frying.

I tried it the other day and the schnitzel came out great. Shaking the pan keeps the oil temperature more even and help form a nice crisp crust.

It may depend on the pan & how well it holds the heat…

I've had the chance recently to test three different frying pans of three vastly different qualities & prices, and the results were amazing. All are approx 30cm [12"] pans, smallest 29, largest 32.

£15 Cheapo lightweight supermarket 'teflon' pan - get the oil nearly smoking before putting in the chicken. First side cooks in seconds, almost too fast. by the time you flip it, the pan has lost all its heat & the second side takes minutes. Lifting the flame for the second side doesn't really help as it then gets hot spots.
Result: crispy one side, soggy the other, or burned at the edges. Poor.

£50 Heavy 'premium brand' - don't need the oil quite so hot to start with. Less pronounced temperature drop by the time you flip it.
Result: Quite acceptable. Been using this one for years & have got used to just easing the flame up a little to compensate for the second side.

£120 Seriously 'posh' pan - takes ages to heat up, but once it does it can weather the storm. Same cooking time both sides of the chicken, at an overall lower starting temperature than either of the other pans. Heat stability quite remarkable.
Result: Best schnitzel I've ever had.

If you want brands, the first is a no-name supermarket's own brand, second is a Meyer, Raymond Blanc, & the expensive one is Scanpan.

The oil should be heated in such a way that the starch coating's surface reaches 190 C during the first few seconds of frying. This is a general rule that holds for all deep fried things with a starchy surface (which is practically all of them, since the nonstarchy ones typically get breaded).

There are various ways to achieve this. The easiest is to use a deep fryer with a sufficient amount of oil (at least 2 liters is good) held at a constant temperature just above the needed 190 C (to counteract the cooling from the schnitzel insides). With some experience, you can make do with other methods as described in other answers.

The main thing to remember is that the momentary temperature of the oil the moment you add the schnitzel is not sufficient to ensure a well-fried schnitzel. You will have to control the whole heating process properly, and depending how you do it, the oil temperature at that specific point in time can vary. The really important parameter is the time it takes to stabilize the crust at the needed 190 C. If you either let it go up to that temperature slowly, or have it overshoot at first and cool down later, you will end up with a greasy, soggy crust.

I am from Germany so we generally make schnitzel with veal or pork but I think these tips will also apply to chicken.

The most important part to watch out for is something @Tetsujin already hinted at. You are going to introduce a lot of cold (or more scientifically accurate remove heat) from the pan when you put in your schnitzels, meaning your first ones might actually have the oil at the correct temperature but the pan not yet be preheated properly.

What I do is generally that we use ghee for frying schnitzel and you want it to cover your meat. I preheat my pan for 5 minutes on low heat (gas stove) and then add the ghee to it while turning up to medium heat. Let the ghee melt and slowly come up to temp for the course of about 5 minutes(might vary depending on the stove). If it is not getting hot enough this is the point where I'll increase to medium-high. The correct temperature is a little bit above frying temperature, so if I want to fry at 170C (which is my preferred temp) (338F) I will heat until 180-185C (356F-363F) and then drop in my meat. After I get out the first one I will leave the pan undisturbed for the ghee to come back up to temperature.

As an Austrian I feel obliged to answer :)

First, schnitzel should be thin. Pound them really quite thin, salt and pepper them, then do the breading and put them in the pan soon after (no "pre-breading").

The oil should be about 1 cm/half an inch high in a wide pan and be about 190 °C/375 °F. I like to use an instant read thermometer to periodically check the temperature, but if you do not have one, you can dip the tip of a breaded schnitzel into the oil; it should immediately start to bubble around the crust and make a crackling sound.

Do not overcrowd the pan, as the temperature will drop, leading to a soggy crust. Agitate the oil slightly from time to time by shaking to keep the temperature uniform.

When a schnitzel is done and has developed a nice golden crust, transfer it out of the pan onto a plate with some kitchen towels or a cooling rack to let excess oil drain of. You can keep a few schnitzels in a low oven until you are ready to serve all of them.