Whisk the eggs with 1/2 teaspoon salt in a large bowl. Heat 1 tablespoon olive oil in a large nonstick skillet over high heat. Add all but 1/2 cup of the mushrooms and season with 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring, until the mushrooms brown, about 5 minutes; remove to a plate.
Wipe out the skillet and melt the butter over medium heat. Add the eggs and cook, stirring with a rubber spatula, until small curds form and the eggs are mostly cooked. Spread any uncooked egg to fill in the gaps and cook, undisturbed, until just set on the bottom, about 5 minutes. Remove from the heat and sprinkle with the cheese. Arrange the cooked mushrooms down the middle and fold in the sides. Slide the omelet onto a cutting board and cut crosswise into 4 pieces. Spread the rolls with the mustard and fill with the omelet.
Whisk the vinegar, tarragon, remaining 1 tablespoon olive oil, 1/4 teaspoon salt and a few grinds of pepper in a large bowl. Add the frisée and reserved mushrooms; toss. Serve with the sandwiches.