For the dough, combine the water, yeast, honey, and olive oil in the bowl of an electric mixer fitted with a dough hook. Add 3 cups flour, then the salt, and mix. While mixing, add 1 more cup of flour, or enough to make a soft dough. Knead the dough on low to medium speed for about 10 minutes until smooth, sprinkling it with flour, if necessary, to keep it from sticking to the bowl.

When the dough is ready, turn it out onto a floured board and knead by hand a dozen times. It should be smooth and elastic. Place the dough in a well-oiled bowl and turn it several times to cover it lightly with oil. Cover the bowl with a kitchen towel. Allow the dough to rest at room temperature for 30 minutes.

Divide the dough into 6 equal parts and roll each one into a smooth ball. Place the balls on a baking sheet and cover them with a damp towel. Allow the dough to rest for 10 minutes. Use immediately, or refrigerate for up to 4 hours.

If you've chilled the dough, take it out of the refrigerator approximately 30 minutes ahead to let it come to room temperature. Roll and stretch each ball into a rough 8-inch circle and place them all on baking sheets sprinkled with cornmeal. (You will be able to fit 2 pizzas on each 18 by 13-inch baking sheet.)

Light your grill and wait until it's hot.

Place the pizzas directly onto the grill and cook on 1 side for 1 minute. Turn the pizzas over and brush with olive oil or garlic oil.

Top the pizzas with any toppings you wish, piling them high. Drizzle each pizza with 1 tablespoon of olive oil. Put the lid on your grill and cook for 5 minutes more, until the crust is crisp and the toppings are cooked.