Heat the oven to 450 degrees F. Lay the garlic on a square of foil. Drizzle 1 teaspoon of the olive oil over the top. Bring up the sides of the foil and wrap up the garlic. Roast until tender, about 40 minutes. Squeeze the garlic pulp out of the skins and into a small bowl; mash the pulp with a fork.

Rub the portabella caps all over with the 1 tablespoon olive oil and sprinkle with salt. Put them, gill side up, in an ovenproof pan. In a small bowl, combine the shallots, chopped garlic, thyme, balsamic vinegar, and salt and pepper to taste. Sprinkle the mixture evenly over the portabellas. Roast (you can do this at the same time as the garlic) until the mushrooms are tender and browned, 20 to 30 minutes. Remove the portabellas from the oven and set the oven temperature to broil.

Toast the bread slices on both sides under the broiler. Spread each slice with some roasted garlic puree; sprinkle with salt and pepper. Top four of the slices with a portabella; put them on a baking sheet. Sprinkle the cheese over the portabellas and return them to the oven to melt the cheese. Top with the parsley and another bread slice to make a sandwich.