In a large skillet, melt butter over medium-high heat. Add mushrooms; cook, stirring frequently, until browned, 5 to 7 minutes. Reduce heat to medium. Add asparagus, 1½ teaspoons salt, ¾ teaspoon pepper, and 2 tablespoons tarragon. Cook, stirring frequently, until softened and fragrant, 3 to 4 minutes. Transfer to a serving platter and top with watercress.

In a small bowl, whisk together lemon juice, garlic, remaining ½ teaspoon salt, remaining ¼ teaspoon pepper, and remaining ½ teaspoon tarragon. Slowly pour in oil, whisking continuously, until fully emulsified. Lightly drizzle over asparagus mixture and serve.