Preheat oven to 350 degrees. Slice the top from the head of garlic, place in an ovenproof bowl and drizzle with olive oil. Cover tightly with foil and roast in the oven for 1 hour. While the garlic is roasting, make the sandwich fillings.
Saute shiitake mushrooms with two tablespoons of butter, salt and pepper in a medium skillet over low heat, stirring frequently, until edges are golden brown. Remove from heat.
Tear the prosciutto into bite-sized pieces and set aside.
Mix the tomatoes, lemon juice and garlic-chili paste in a small bowl, coating well. Drain in a colander.
When the garlic is done roasting, squeeze out the cloves and mash with a fork. Spread equal amounts of roasted garlic on 6 of the bread slices.
Spread one tablespoon of Dijon mustard on each of the remaining six bread slices. On each of these slices, add equal amounts of cheese. Top with equal amounts of arugula, tomatoes, mushrooms and prosciutto. Top with the garlic bread slices.
Heat remaining butter in a large skillet or grill pan on medium-high heat. Toast sandwiches, covered, for two-to three minutes per side, or until golden brown. If you have to work in batches, keep first batch of sandwiches warm in the oven on low heat. When all sandwiches are done, slice on the diagonal and serve.
*Sriracha, or garlic-chili paste, can be found in the Asian section of any supermarket