Melt butter in a small nonstick skillet over medium heat. Add mushrooms; cook, stirring often, until tender and lightly browned, 5 to 7 minutes. Add garlic; cook, stirring constantly, until fragrant, about 1 minute.

Meanwhile, whisk eggs, milk, tarragon, salt and pepper in a small bowl. Once the mushrooms are tender, stir the egg mixture into the pan. Cook, tilting the pan to let the egg run around the edges, until the egg forms a thin, even layer. Continue to cook, reducing the heat if browning too quickly, until just slightly wet, 1 to 2 minutes. Sprinkle with cheese. Use a spatula to roll up the omelet. Transfer to a plate; sprinkle with chives.