1. Whisk together vinegar and mustard, with salt and pepper to taste. Add oil in a slow stream until emulsified.
2. Arrange mushroom caps stem-side up on a plate and drizzle with half of vinaigrette. Season with salt and pepper to taste.
3. Preheat grill. Grill mushroom caps on a lightly oiled rack, stem-side up, uncovered, over moderately high heat until underside is softened and golden brown, about 5 minutes. Turn caps over and grill until just tender, about 4 minutes more. Turn caps over and top each with a 1-ounce round of goat cheese. Continue to grill, spreading goat cheese slightly with a spatula, until cheese is slightly melted, about 1 minute. Transfer caps to serving plates.
4. Put arugula in a bowl and toss with enough vinaigrette to coat. Mound salad on top of mushroom and sprinkle with tomatoes. Drizzle any remaining vinaigrette over and around mushroom.