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Cook shallot in 2 Tbsp butter with 1/4 tsp salt and 1/8 tsp pepper in a 10-inch nonstick skillet over medium heat, stirring occasionally, until softened, 3 to 4 minutes.
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Stir in mushroom stems and cook, stirring often, until tender, about 5 minutes. Stir in tarragon and transfer to a bowl.
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Beat eggs with 1/4 tsp salt and 1/8 tsp pepper until well combined. Heat remaining Tbsp butter in same skillet over medium-high heat until foam subsides, then add eggs and cook until underside is set, about 1 minute. With a fork, pull set eggs to center, letting uncooked eggs run underneath. Before eggs are completely set, add mushroom mixture and cheese to one half, on side away from handle. Fold other half of eggs over filling with a heatproof rubber spatula, tilting pan as you roll omelet onto a plate.
