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.
Master directions for grilling pizza For Gas Grills Heat the grill by setting all the burners on high. After lighting, close the lid and leave on high for 10 minutes, then reduce the heat of all the burners to medium.
Pick up the oiled dough by the two corners closest to you. In one motion, lay it down flat on the cooking grate from back to front (as you would set a tablecloth down on a table). Close the lid and grill for 3 minutes (no peeking!), then check the crust and, if necessary, continue grilling a few more minutes until the bottom is well marked and nicely browned.
Use tongs to transfer the crust from the grill to a peel or rimless baking sheet. Close the lid of the grill. Flip the crust to reveal the grilled side. Follow the specific recipe directions for adding any sauce, toppings, and/or cheese.
Switch the grill to indirect heat by turning off the center burner(s) if you have a three- or four-burner grill. For a two-burner grill, turn off one burner.
Set the pizza back on the grate over indirect heat (the unlit section) and grill, with the lid down, until the bottom is well browned and the cheese is melted, 7 to 10 minutes. For two-burner grills, rotate the pizza halfway through the cooking time.
Remove from the grill, garnish, and season as directed. Slice and serve immediately.
For Charcoal Grills Build a fire by lighting 50 to 60 charcoal briquettes in either a chimney starter or in a pyramid-shaped mound on the bottom grate of your grill. Once the briquettes have become gray-ashed (20 to 30 minutes or 450°F inside the grill), move them all to one side of the grill.
Pick up the oiled dough by the two corners closest to you. In one motion, lay it down flat—over the side without briquettes—on the cooking grate from back to front (as you would set a tablecloth down on a table). Close the lid and grill for 3 minutes (no peeking!), then rotate the crust 180 degrees and continue grilling until the bottom is well marked and evenly browned, another 2 to 3 minutes.
Use tongs to transfer the crust from the grill to a peel or rimless baking sheet. Close the lid of the grill. Flip the crust to reveal the grilled side. Follow the specific recipe directions for adding any sauce, toppings, and/or cheese.
Set the pizza back on the grate over the side without briquettes and grill, with the lid down, for 4 to 5 minutes. Rotate the pizza 180 degrees and continue to grill with the lid down until the bottom is well browned and cheese is melted, another 4 to 5 minutes.
Remove from the grill, garnish, and season as directed. Slice and serve immediately.
