Mix and knead together all of the dough ingredients -- by hand, mixer or bread machine -- till you've created a smooth, soft dough. Don't over-knead the dough; it should hold together, but can still look fairly rough on the surface.
Allow the dough to rise, covered, for 45 minutes; then refrigerate it for 4 hours (or up to 36 hours); this step will develop the crust's flavor.
Clean the scallions, then cut off and discard the top half of each. Use a small, sharp knife to cut short strips about 1 to 2 inches long at the bottom (bulb end) of the scallion without cutting all the way through to the center; place the scallions in ic
Slice the tomatoes 1/4-inch thick; oven-dry them by placing them in a single layer onto greased or parchment-lined sheet pans, and baking them in a preheated 400°F oven for 20 to 30 minutes; turn the oven off and let the tomatoes cool slowly. Clean and sl
Drain the scallions, then lightly brush the scallions and mushrooms with olive oil (or use a mister or sprayer). Grill the scallions and mushrooms until they're soft and brown around the edges. Remove them from the grill and set them aside (in a single la
About 45 minutes before you want to grill the pizza, remove the dough from the refrigerator, and shape it into a 14- to 16-inch round. Place it on a piece of parchment paper, or on a baker's disk. Preheat your grill to medium temperature.
The easiest way to grill pizza is to use a baker's peel to transfer the crust from the counter to the grill. Pick up the risen crust and baker's disk, and place the pizza and disk onto the grill. If the pizza is on parchment, carefully lift it off the par
Place the scallions, tomatoes, and mushrooms atop the pizza, then sprinkle with the grated cheese and Pizza Lovers' Seasoning. Close the grill cover just long enough to melt the cheese, then remove the pizza from the grill. Serve it immediately.