In a medium bowl, combine the mushrooms with the olive oil, shallots, rosemary, and thyme; let sit for about 10 minutes.
Light a fire in a charcoal or gas grill. Using tongs, remove the mushrooms from the marinade and grill until slightly softened and browned, about 2 minutes on each side. Set them aside. (You can also complete this step ahead of time, refrigerate the mushrooms, and then bring them to room temperature when you’re ready to stuff the tomatoes.)
Meanwhile, prepare the tomatoes so that you can use them to serve your salad in.
Preheat the oven to 325º F. Using a sharp knife, slice the tops off the tomatoes either flat or in a zigzag pattern; you need them later so don’t discard them. Using a melon baller or spoon, scoop out the insides. (You may want to save the insides for a sauce or another salad.) Set aside. Place the tomatoes in a 1-inch-deep baking pan (you may have to adjust the bottoms—cutting carefully and only slightly—if they can’t sit upright), with their tops. Bake until the skin starts to wrinkle, about 15 minutes.
When the tomatoes are finished, remove the tops, and stuff with alternate layers of mushroom mixture and cheese. Place the tomatoes back in the oven (without tops) just until the cheese starts to melt, about 5 minutes.
Top each tomato with a bit of the mixed baby greens and a drizzle of the vinaigrette. Cover with the tomato top, and serve.