Whisk together the yeast and 1/4 cup of the water in a small bowl and let sit for 5 minutes to proof. Place 1 3/4 cups of the flour in a large bowl with the salt. Add the yeast mixture, the remaining water, and oil and stir until combined. Transfer the dough to a floured surface and knead until smooth. Grease a large bowl with the remaining oil, add the dough and turn to coat. Cover and place in a warm place until doubled in volume, about 1 to 1 1/2 hours.
Preheat the grill to high. Once the dough has risen, divide it in half, and divide each half into 2 balls. Roll each ball into an 8-inch circle. Brush with oil and grill on both sides until golden brown.
Spread about 1/2 cup of the sauce on each pizza and top with goat cheese and basil. Place pizza on the grill and cook until bottom crust is golden brown and cheese is bubbly. Remove from heat, drizzle with red chili oil, and serve.
Tomato-Red Pepper Sauce:
Heat oil in a medium saucepan over medium heat. Add onion and garlic and cook until soft, about 3 to 4 minutes. Add red peppers and tomatoes and cook until thickened. Season with salt and pepper, to taste. Puree in a food processor until smooth. Let cool slightly.
Red Chili Oil:
In a blender, combine oil, red chili flakes, and salt; blend for 1 minute. Strain into a small bowl or squeeze bottle.