1 tablespoon finely chopped shallot (from about 1/2 medium shallot)
4 ounces white button mushrooms (about 6 medium), cleaned, stemmed, and sliced 1/4 inch thick
1/4 teaspoon finely chopped fresh thyme leaves
2 teaspoonsdry sherry
4 teaspoonswhole milk
1/2 teaspoonkosher salt, plus more as needed
1/4 teaspoonfreshly ground black pepper, plus more as needed
1/4 cup shredded fontina cheese (about 1 ounce)
Melt 1 tablespoon of the butter in an 8-inch nonstick frying pan over medium-high heat until foaming. Add the shallot and cook, stirring occasionally, until softened, about 2 minutes.
Add the mushrooms and thyme, season with a pinch each of salt and pepper, and cook, stirring rarely, until the mushrooms are golden brown, about 4 to 5 minutes. Add the sherry and cook until evaporated, about 1 minute. Transfer to small heatproof bowl and set aside. Wipe out the frying pan with a paper towel; set the pan aside.
Whisk the eggs, milk, and measured salt and pepper in a large bowl until pale yellow and the egg yolks and whites are evenly combined. Set 2 serving plates aside.
Melt 1 tablespoon of the butter in the reserved frying pan over medium heat until foaming. Add half of the egg mixture (about 3/4 cup) to the pan and stir constantly with a rubber spatula, moving the eggs around until they form small curds, about 2 to 3 minutes.
Gently shake the pan and use the spatula to spread the egg mixture evenly across the pan—the top of the eggs should have a creamy consistency. Sprinkle with half of the cheese. Spread half of the mushroom mixture down the middle third of the egg mixture.
Remove the pan from heat. Using the spatula, fold a third of the omelet over and onto itself, making sure to fold it over the filling. Gently push the folded side of the omelet toward the edge of the pan. Tilt the pan over one of the serving plates and roll the omelet onto the plate, seam side down.
Repeat with the remaining tablespoon of butter, eggs, fontina, and mushroom filling. Serve immediately.