Preheat oven to 375°F. Cover stack of phyllo with 2 overlapping sheets of plastic wrap and a dampened kitchen towel. Put 1 phyllo sheet on a work surface and brush with some butter, then top with 2 more sheets of phyllo, brushing each with butter.
Cut buttered stack into 6 (5-inch) rounds using with a round cutter. Line each of 6 muffin cups with a round. Make 6 more phyllo cups in same manner with remaining pastry sheets and butter. Bake cups about 8 minutes, then cool completely in pan on a rack.
Melt butter in a 3-quart heavy saucepan over moderately low heat, then whisk in flour. Cook roux, whisking, 3 minutes. Add milk in a stream, whisking, and bring to a boil, whisking. Reduce heat and simmer, whisking occasionally, 5 minutes.
Remove from heat and whisk in mustard, egg yolks, and 1/4 cup Parmesan until combined, then fold in goat cheese.
Beat egg whites with cream of tartar in a large bowl with an electric mixer until they just hold stiff peaks. Fold one third of whites into sauce to lighten, then fold in remaining whites gently but thoroughly.
Spoon batter into 8 phyllo cups and sprinkle with remaining Parmesan. Bake until soufflés are puffed and golden, about 15 minutes.
Make salad while soufflés bake:
Whisk together vinegar, mustard, and salt in a bowl, then add oil in a slow stream, whisking until emulsified. Just before soufflés are ready, toss mache and endives in a large bowl with just enough dressing to coat.
Mound salad onto 8 plates and sprinkle with chives, then make a small nest in center of each. Place a soufflé cup in each salad and serve immediately.