Preheat the grill or broiler. Mound the flour and salt in the center of a cutting board. Using a pastry cutter, cut the butter into the flour until the mixture resembles coarse crumbs. Mix in the egg and cold water and work the dough just enough so that it comes together. Wrap in plastic and refrigerate 30 minutes.
Brush the eggplant slices with olive oil and grill on both sides until marked and crispy, about 3 minutes per side. Set aside. Meanwhile, in a 12 to 14-inch skillet, heat the olive oil over high heat until just smoking. Add the garlic and onion and saute over high heat until light golden brown. Add the mint, thyme, oregano and salt and pepper to taste and cook 5 minutes more. Stir in the tomato puree, bring to a boil, simmer until very thick and stew like and remove from heat.
Preheat the oven to 350 degrees F. Roll the dough out to an even 1/8-inch thickness and fit it into a 10-inch tart pan. Blind bake the pastry shell for 20 minutes, until golden brown, then remove from oven and allow to cool. Layer the eggplant slices and tomato sauce mixture in the tart shell, alternating layers and ending with a layer of eggplant. Cut the slices if necessary to fit the pie shell. Top with abundant grated cheese and bake in the oven 20 minutes, until the cheese is browned and the sauce is bubbling. Let cool 15 minutes, then slice and serve.