What is the most energy efficient way to cook meat on an electric stove?

As @Jefromi says, there is no simple answer. Also depends on your definition of "cooked".

1. For water vs oil, most likely oil is more energy efficient. Frying a 16 oz steak takes only ~5/10 minutes, whereas boiling 16 oz of meat in water will take at least double that time, even if you use exact same pan and exact same stove. If you cut the meat more strategically, you can save some energy, which brings me to 2nd point.

2. The shape of the meat will have a big influence. For example, a 1 inch thick tenderloin will take considerably more energy to "cook" compared to 4 tenderloin steaks of quarter inch, assuming you can lay all 4 steaks in a pan together.

There is a saying that in ancient China, stove fuel was hard to come by, thats why they always used to cut the meat as thin as possible so that the dish finishes cooking fast with minimum stove-time. Don't know if its historically accurate, though :)

Answering more as a chemical engineer - we study heat transfer.

A lid clearly reduces heat loss. A lid also turns the pan into an oven - you hold the heat and use it on the non burner side.

A little bit of water will put more molecules in the vapor phase for more heat transfer but dilutes flavor. More than little bit is waste. Steam is 1000 x the volume of water. My mom will steam sausage links to cook them a fast but I think it kills the flavor.

Oil increases heat transfer on hot side but too much oil just makes it taste greasy. It basically increases the contact area. Oil does not vaporize nearly to the extent of water so not doing much on the non heat side.

A pressure cooker on low to medium heat is going to be most efficient. A maximum amount of heat stays in the vessel. Let it build some heat and then turn off the heat and let it finish. You don't concentrate flavor. What goes in comes out.

I doubt there's a simple perfect answer. You'd want to cook it hot, but not too hot, and with a little bit of water water, but not too much. If you cook too hot you're wasting energy, since it takes time for the heat to propagate to the center of the meat. If you don't cook hot enough to at least keep everything at 100C, you'll be wasting more energy over time. If you don't have any water, then even with oil, you'll be cooking mostly just from the pan surface instead of all around thanks to the steam, while if you have too much you'll waste energy heating it.

So add enough water to fill the pan with steam until it's done cooking, perhaps half a centimeter or so, then heat it as fast as possible to get it completely hot under the lid, then reduce heat but leave it hot enough to stay like that. Might not be perfectly optimal but should be pretty good.

If on the other hand you had ground meat, you can cook it as hot as you want, and probably don't need to add any water since it'll release some. As long as you spread it out well, it'll cook much, much faster than a single piece of meat with the same weight.

One consideration that hasn't been mentioned in your selection of a pan. If you use something like cast iron, you require some time to heat it up, but it will continue to cook the meat even after you've turned off the burner.

A thinner pan will require leaving the burner on the whole cooking time.

Personally, I'd look into slicing the meat thinly so that you make the most efficient use of the surface area of the pan. Normally, I'd sayd that you'd want a little bit of space between pieces (1cm, 3/8") so that moisture will evaporate and you won't end up steaming the pieces too much, but in your particular case, vented steam is lost energy.

Personally, I'd probably consider cooking in the oven, under the broiler -- you're heating from the top, so the heat goes into the meat first, not the pan. Normally there's a little bit of pre-heating time, as you want the elements to glow red to get a good sear, but there's nothing preventing you from starting the meat in the oven when you're heating the elements.

If you have the opportunity to swap our your stove, you should consider getting an induction burner -- they're much more efficient in transferring heat to the pan than resistive electric burners.