For the dough: Put the water and sugar in the bowl of a stand mixer and sprinkle the yeast on top; let rest until the mixture bubbles, about 5 to 10 minutes. Meanwhile, whisk together the flour and salt in a large bowl to break up any lumps. When the yeast is ready, add the flour mixture and olive oil and mix, using dough hook, on the lowest speed until the dough looks shredded, about 1 to 2 minutes. Increase the speed to medium and continue to mix until the dough is smooth and very elastic, (you should be able to stretch it 2 to 3 inches without breaking), about 6 to 10 minutes.
Turn the dough out of the mixer, form it into a ball, and put into a large, oiled mixing bowl. Turn to coat the dough in the oil then cover with a clean, damp dishtowel, and let rest in a warm place until it doubles in size, about 30 to 45 minutes. (Alternatively, refrigerate the dough and let rise 12 to 24 hours before using.) Punch the dough down and shape as desired.
For the zucchini: Heat the grill to medium-high (about 375 degrees F) and rub the grate with a towel dipped in oil. Add 2 tablespoons olive oil and the zucchini to a large bowl and toss to coat. Grill the zucchini, covered, turning rarely, until tender and charred, 5 to 7 minutes.
Transfer the cooked zucchini to a bowl and add the red wine vinegar, and red pepper flakes. Season with the salt and pepper, to taste, and toss to coat. Set aside until ready to use.
For the pizza: Heat the grill to medium-high (about 375 degrees F) and rub the grate with a towel dipped in oil.
Oil your hands, put half of the dough (about 1 pound) on a piece of parchment paper, and press the dough into a rectangle about 1/4-inch thick. Drizzle the dough with oil. Lay the dough, oiled side down, on the grill and remove the parchment paper. Grill the dough until golden brown, toasted, and grill marks form on the bottom, about 2 to 3 minutes. Flip the dough, brush with 1 tablespoon oil and top with half of the mozzarella. Cover with half of the zucchini and half of the goat cheese. Close the grill, checking and rotating as needed to avoid burning, until dough is lightly charred and cheese is melted, about 8 to 10 minutes.
Transfer the pizza to a cutting board, top with half of the basil leaves, and half of the sun-dried tomatoes, and serve. Repeat to make a second pizza.