First, make the mushroom duxelles. Very finely chop the mushrooms and shallots, or pulse in a food processor to a fine consistency.
Heat 2 tablespoons unsalted butter in a wide, deep skillet over medium-high heat until foamy. Add the mushrooms and shallots and spread evenly over the surface of the pan. Allow to cook 3 minutes, undisturbed, then season generously with salt and stir. Allow to continue cooking, stirring every 2 minutes or so with a wooden spoon or spatula until the mushrooms have released their liquid and are beginning to stick to the bottom of the pan.
Add the cream and stir to combine. Turn the heat down to medium and allow to simmer for 2 minutes. Stir in the parsley. Remove from heat and set aside.
This will make more mushroom duxelles than is needed for the omelettes; it will keep for several days in the refrigerator in an airtight container.
Step 2
Next, make the omelettes. In a small bowl, whisk 2 eggs to a smooth consistency.
In a small nonstick skillet, heat ½ tablespoon butter over medium-high heat until foamy. Pour the eggs into the pan, scraping the bowl with a silicone spatula. Immediately turn the heat to medium-low. As the sides begin to set, loosen and lift up with a wooden spatula, turning the pan to allow the uncooked eggs to set. This will only take about 2 minutes. Turn the heat to low.
Sprinkle 1-2 tablespoons cheese down the center of the omelette. Spoon 2-3 tablespoons mushroom duxelles over the cheese, evening out with a silicone spatula as needed. Top with 1 tablespoon fresh chives.
Tip the pan and fold ⅓ of the omelette toward the center, then fold the remaining third. Slide out of the pan and onto the serving plate. Repeat for the second omelette. Serve immediately.