Combine the flour, grated Parmesan, and 1/4 teaspoon of salt into a large mixing bowl. Incorporate the butter and shortening into the dough, using a pastry blender or a fork, until pea-sized pieces form. Add the water, 1 tablespoon at a time, just until the dough comes together, being careful not to over mix.
Form the dough into a disk, wrap tightly in plastic wrap, and place in the refrigerator to rest for at least 30 minutes.
Preheat oven to 450 degrees F.
Remove the dough from the refrigerator and place on a lightly floured surface. Roll dough into a 12-inch circle. Fold dough in half and place into a 10-inch fluted tart pan. Unfold dough and press gently into the bottom and sides of the tart pan, trimming off excess dough as needed. Line the dough with parchment paper and fill with pie weights. Bake crust for 18 to 20 minutes, or until lightly browned. Remove the crust from the oven and brush with the egg white wash while still warm. Set aside to cool.
Reduce oven temperature to 350 degrees F.
Slice the tomatoes into 1/4-inch slices and lightly season on both sides with salt and pepper and set aside. Spread the garlic puree evenly over the bottom of the tart crust. Sprinkle the bread crumbs evenly over the garlic puree. Arrange the green onions, tomatoes, and chopped basil in the tart pan. Sprinkle the cooked bacon over the tops of the tomatoes. In a small bowl, combine 1/4 cup of the Parmesan with the mayonnaise and spread evenly over the top of the tart. Sprinkle the remaining cheese over the top and bake until the tomatoes are wilted and the top of the tart is golden brown, 35 to 40 minutes.
Remove from the oven and cool on a wire cooling rack for 10 minutes. Remove the tart pan ring and cut into wedges. Serve tart on a bed of tomato water, if desired. Garnish with fresh basil sprigs and serve immediately.
Preheat the oven to 325 degrees F. Line a baking sheet with parchment paper or aluminum foil.
Cut the top quarter off each garlic head so the cloves are exposed. Place cut side up on the prepared baking sheet. Drizzle the oil over the garlic and season lightly with the salt and pepper. Turn the garlic cut side down. Roast until the cloves are soft and golden brown, 1 to 1 1/4 hours.
Let the garlic cool. Squeeze the soft garlic flesh from each head into a bowl. Mash the flesh with a fork to blend it into a puree. (The puree can be stored in an airtight container in the refrigerator for up to 2 weeks.)
Combine all ingredients in a blender or food processor. Add just enough olive oil for flavor and season with salt and pepper, to taste.