Crust:
1 cup warm water
1 Tbsp. sugar
2 1/4 tsp. active dry yeast (one 1/4 ounce package)
2 1/4 cups all-purpose flour
1 cup cake flour
1 Tbsp. kosher salt
2 Tbsp. olive oil
Combine water, sugar, and yeast. Proof until mixture turns foamy, about 5 minutes.
Mix flours and salt in the bowl of a stand mixer fitted with a dough hook.
Add oil to proofed yeast mixture, then pour into flour mixture. Knead dough on low speed for 10 minutes. (If kneading by hand, knead dough for the same amount of time.)
Place dough in a lightly oiled bowl, turning dough to coat surface. Cover dough with plastic wrap and let rise in a warm place until doubled, about 2 hours.
Punch dough down and divide it into four balls, pinching the bottoms closed. Cover with plastic wrap, and let rise in a warm place another hour.
Grilling the pizza:
Roll out the dough on a dry work surface dusted with flour. Shape it into a 10- to 12-inch circle or rectangle.
Preheat one side of the grill to medium-high, the other side to low. Place dough over hot side, close lid, and let cook 2-3 minutes. The crust will bubble and grill marks will appear underneath.
Use tongs to flip the crust over and move it to the low heat side of the grill. Brush the crust with olive oil – it adds flavor and contributes to the crispy texture.
Add toppings. Keep quantities light so they cook quickly. Return pizza to the hot side of the grill to finish cooking.
Pizza is done when the cheese is melted. Remove pizza from the grill using a pizza peel or the back of a baking sheet.