Heat 2 teaspoons butter in a medium skillet or sauté pan until foaming, then add the mushrooms. Cook until brown and caramelized, 5 to 7 minutes. Season with salt and pepper and remove to a bowl or plate. Wipe the pan clean and return to the heat. Add the prosciutto slices (don't crowd) and cook for a minute or two, until the pieces begin to shrink slightly and crisp up. Reserve. Heat 1 teaspoon butter in a small nonstick pan set over medium heat. Beat 2 eggs with 1 tablespoon milk and season with salt and pepper. Add the eggs to the pan and use a wooden spoon or heatproof rubber spatula to move them about, as if you were scrambling them. Continue to do this for 30 seconds or so, until about half of the eggs have set, then use your spoon to gently lift the edge of the omelet and swirl the liquid egg around so that it runs underneath to the pan. When all but the thinnest film of egg has set, add 1⁄4 cup cheese and a big spoonful of the sautéed mushrooms. Fold the omelet over (either once for a half-moon or twice for a long thin omelet) and gently slide onto a warm plate. Garnish with crispy prosciutto and chives (if using). Repeat to make 4 omelets in all.