My new pan has a max temperature of 350 F but the recipe calls for 425 F. What should I do?

The best solution is - use a different pan if you want the recipe's advertised results.

If something is baking at 425 F for only 15 minutes, it's very likely that you need that high heat to get the result you want - which is generally going to give you a toasty, well-browned exterior and a cool, just done interior.

If getting an oven-safe pan isn't possible, you can certainly change your oven temperature to 350 F but be aware that, depending on your oven, it may not be properly calibrated¹ and it may actually be hotter than 350 F, which means you may be damaging your pan or exposing your food to fumes that the plastics in it are giving off when being overheated. So, even then, I'd be hesitant to recommend cooking it at 350 unless you have an oven thermometer and know your oven heats to the correct temperature.

So, my recommendation would be to go even lower, 325 F or so and then start at your original cooking time (15 minutes). If this is some sort of meat (including fish) use a probe thermometer to check the internal temperature every few minutes after that. Depending on the readings you get, you can go for more or less time... if you're half-way to your target internal temperature after 15 minutes, you can probably go about 5-7 minutes more before checking but as you approach the temperature, you're going to want to wait only a minute or two.

If this is some sort of baked good (cake, cookie, etc), I'm more concerned that you won't like the outcome at this lower temperature but you might get lucky. Check it after 15 minutes and then every couple of minutes until it reaches the done test described in the recipe, whether that's poking it with a toothpick or golden color etc.

I can't give you one number to aim for because time is an incredibly poor way to cook anything, so you shouldn't be using time to cook anyway. Always cook based on the doneness cues mentioned in the recipe. If your recipe doesn't include these non-temporal cues, you're probably better off finding a new recipe.

For a scientific discussion of this, I recommend you read the excellent answer to this related question.

1. Note, even a properly-calibrated oven will still heat to a temperature around your target because it cycles on and off. So, it will likely overheat slightly, turn off the heating element until the temperature drops below the target and then heat back up again.

First and foremost I hope your pan is oven safe (all parts can be heated and won't melt in ex. plastic). To be 100 % sure read your manual or there might be a specification on the pan itself. As for the time you need to cook it, it will always vary so the universal rule of cooking is to cook until it's done. You should never follow a recipe blindly (your slices might be thicker, oven weaker, etc), instead just check your dish in 6-7 minute intervals once you think it's near cooked. Hope this helps :)

Assuming the temperature limitation is caused by some plastic parts on the handle, take one or two sheets of tinfoil, wrap the handle with it and proceed with your recipe and the original temperature instructions.

I've done this uncounted times, works fine and my pans don't suffer from the oven heat.

However, if the limitation is not caused by some plastic parts but the pan itself: throw it out and get yourself some real tools. 425 F (220 °C) are not extraordinarily high, so your cookware should easily withstand them.