In a large bowl (the bowl of your mixer, if you have one), combine water, sugar, and yeast. Let stand for 10 minutes or until yeast is bubbly. Add salt and stir. Add 1 1/2 cups of flour and mix well. Gradually add more flour (usually between 3-4 cups, depending on your elevation and your humidity) until dough starts to pull away from the sides of the bowl and it barely sticks to your finger. Cover and allow to rise for 45 minutes or until doubled in bulk. Remove from bowl and place on a surface sprayed with cooking spray. Punch down, knead slightly, and place back in bowl. Cover bowl and allow dough to rise for another 30 minutes. Prepare sauce.
Sauce
Empty tomato paste into a bowl and add water a few tablespoons at a time, stirring constantly until combined. Add remaining ingredients, stir to combine, and allow to stand until ready to use.
Roll Out the Dough
Divide the dough into 4 balls. Roll out each ball as thin as you can – not any thicker than 1/4″ inch, but thinner if you can. When your dough is flattened out, drizzle a little olive oil on one side and brush it around. If you’d like, sprinkle it with a little garlic or onion salt.
Preheat Grill
First, you need to preheat your grill. This is an important step. The rack needs to be nice and hot so the dough will start cooking immediately and the overall temperature needs to be hot in order to act like an oven when the lid is closed. Set grill to medium-high heat and shut the lid. While the grill is pre-heating, prepare your other ingredients.
Prepare Toppings
You have to move very quickly while preparing these pizzas, so set up a platter with the toppings listed (bell pepper, olives, shredded cheese, and so on) and keep it on hand next to the grill.
Grill
Place the dough, oiled side down, onto the hot grill and then shut the lid immediately. You’ll want the heat to stay inside so it acts like an oven and cooks the dough. The timing all depends on the heat of the grill and the thickness of the dough, but it usually only takes about 5 minutes or less. Have a spatula handy to lift up the dough and check it. You’ll want to see nice brown grill marks, but avoid burning it. If you can tell that your dough is cooking too fast and too hot, you may want to use indirect heat. For a gas grill with multiple burners, turn off the burner directly under the pizza, but leave the others on. For a charcoal grill, move the charcoal to one side of the grill and place the pizza on the other side.
These only need a few minutes - check the bottom for nice grill marks and make sure it’s dry to the touch before flipping. Right before you flip it, brush a little olive oil on the uncooked side.
After you flip the crust, brush lightly with pizza sauce and then spread some of the olive pesto over the sauce. Top with the colored bell peppers, sliced red onions, mozzarella, and feta. When the pizza is cooked through and the cheese is melted, remove from the grill and top with the fresh baby spinach. Repeat with the remaining dough (you can probably cook 2 pizzas at once on an average-size grill).