Heat oven to 400° F. Unfold the pastry and cut it into 4 squares. Place the squares on a parchment-lined baking sheet and, using a fork, prick the dough all over.
Meanwhile, heat 2 tablespoons of the oil in a large skillet over medium-high heat. Add the mushrooms and shallot and season with ½ teaspoon salt and ¼ teaspoon pepper. Cook, tossing occasionally, until the mushrooms are browned and tender, 4 to 5 minutes. Add the wine and cook until nearly evaporated, about 1 minute more.
Dividing evenly, sprinkle the pastry squares with half the cheese, leaving a ½-inch border on each. Top with the mushroom mixture and remaining cheese. Bake until the pastry is golden brown, 20 to 25 minutes.
In a large bowl, whisk together the mustard, vinegar, the remaining 2 tablespoons of oil, and ¼ teaspoon each salt and pepper. Add the greens and radishes and toss to coat. Serve the tarts with the salad and sprinkle with the chives.