Preheat the broiler, with the rack set about 4 inches from the heat source.
In a medium (10-inch) broiler-proof skillet, preferably cast iron, heat the oil over medium heat. Add the onion, season with salt and pepper, and cook, stirring occasionally until the onions are nicely browned and quite soft, about 10 minutes. Remove the onions to a plate.
Melt the butter in the same skillet and add the mushrooms, seasoning with salt and pepper. Turn the heat to medium high and sauté for about 8 minutes, until there is no liquid left in the pan and the mushrooms are beginning to brown.
Meanwhile, in a medium bowl, whisk the eggs, and season with salt and pepper. Return the onions to the pan and add the noodles and thyme. Stir to combine, pour in the eggs, and stir to blend everything evenly. Lower the heat to medium-low and with a spatula, gently lift the edges of the frittata so that the uncooked eggs have the chance to run underneath the noodle-vegetable mixture. Do this every couple of minutes until the frittata is pretty set on the bottom, but the top and middle are still a bit runny.
Sprinkle the feta cheese on top of the frittata and place the skillet under the broiler. Broil until the frittata is set, and the cheese is melted and the whole top is lightly golden, 2 to 4 minutes. Remove from the oven and let sit for a minute or two, then serve in wedges. DON’T FORGET THAT THE HANDLE OF THE PAN IS HOT! You can leave a dishtowel draped over it to remind yourself. Again, also great at room temp or even cold.
Notes
Frittata is delicious leftover, whether hot or cold!