I want to make Tortilla de Patatas but I have extra carrots that I would like to use somewhere. Can I include them in the frying phase? And can the dish be cooked in a casserole dish in the oven instead to make a very tall 'omelet'?
If you would like to add carrots, make sure you cut them small enough (since it takes quite a while to get them soft). I'm not 100% sure it would be a good fit, but that's up to you.
You certainly can make a tortilla in the oven, I've done so in a spring form. I think I did it for about 45 min in an oven of 180°C/350°F. But this depends of course on how much food you use and how big your form is. Just keep an eye on it, the egg should be fully cooked and the potatoes soft.
Tortilla de patatas can be mixed with some other vegetables in order to make some varieties (for example: tortilla campera). But I've never seen a tortilla with carrots, although I understand you need to cook somewhere those carrots. I'd love to know how it looks and tastes! ;)
If you add carrots, make sure carrots are at least a bit cooked before adding to any phase of the tortilla. If I were you, I would take one of these 2 options:
You can make it in the oven if you find it more convenient, why not? Just so you know that is far from a spanish traditional way of cooking it ;)
In case your concern is about making a taller tortilla, you can do that in a frying pan too (no need for an oven to do that). It only takes some more practice to learn until you do a taller one. Why? Basically because it implies that you have to fill much more your frying pan with more tortilla mixture. Which means when you would flip tortilla upside down you may find it heavier than other times and also you have to move quicker too because mixture tends to spread if it's too tall. Also, you should let tortilla in the heat for sometime longer too in order the inside is properly cooked (bearing in mind that the heat can't be very high as the outside could burn).
Well, I hope this help and good luck with your carrot-tortilla! :)