In a large nonstick skillet or cast iron skillet over medium heat, warm 1 tablespoon of the oil. Add the chanterelles whole, sprinkle with half the thyme, and season with salt and pepper. Cook, stirring occasionally, for about 3 minutes.
Add the garlic and stir well. Cook until the mushrooms have shrunk down by half, 5 to 10 minutes more, adding more oil if needed. Transfer the mushrooms to a bowl.
In a separate bowl, mix the eggs with the grated Gruyère and remaining thyme.
In the same skillet over medium heat, melt half the butter. Pour half the egg mixture into the skillet and, using a wooden spoon, swirl it around the pan. As the egg begins to cook and form a layer on the bottom, loosen the omelet with a spatula.
Once it’s free from the bottom of the pan, start dotting half the mushrooms over half the eggs. When the egg on the surface is almost cooked, carefully fold over the mushroom-free side with a spatula to create a half circle.
Gently slide the omelet onto a plate and keep warm while repeating with the remaining ingredients to make a second omelet.
Sprinkle with some extra cheese on top, if desired. Sit down and devour immediately.