Heat a large sauté pan over medium heat for 1 to 2 minutes. Add the butter and 2 Tbs. of the oil. When the butter bubbles, add a pinch of salt, stir, and add the leek. Cook for 2 to 3 minutes, then add the mushrooms and cook, stirring occasionally, until browned on the edges, about 10 minutes. Add the cognac, let sit for 5 seconds, then light a long match to it. Stand back and keep wayward clothing and shaggy hair away from the dancing flames. (If you have a gas stove, be aware that spattering particles may cause the alcohol to ignite prematurely.) The flame should burn out after about 10 seconds. If it continues to burn, put it out by placing a lid on the pan. Remove from the heat and reserve for topping.
Heat your grill as directed in the master instructions below.
Sprinkle your work surface with the cornmeal. Place the dough in the middle of the surface. You can either roll out the dough with a rolling pin, stretch it out with your hands, or press it out from the center against the work surface. Ideally, you want a 12-inch, organically shaped piece of dough—round, square, or rectangular—1/8- to 1/4 inch thick. Just before grilling, drizzle or brush both sides generously with the remaining olive oil.
Grill the first side of the crust following the instructions below. Use tongs to transfer it from the grill to a peel or rimless baking sheet.
Flip the crust to reveal the grilled side. Spread the entire surface with the garlic paste, then with the mushroom-leek mixture. Artfully arrange the cheese over the top. Finish grilling the pizza following the instructions below.
Remove from the grill, sprinkle with the thyme, and season generously with salt and pepper. Slice and serve immediately.
Shave fresh black truffle over the top of the finished pizza.
The delicate, earthy tones of the mushroom medley are a natural for the barnyardy aromas (and that's a good thing!) of a red burgundy.