Whisk the water, sugar, and yeast together in a small cup or container. Set it aside for 5 to 10 minutes, until it becomes frothy. You can move on to the next step in the meantime.
Stir the bread flour, all-purpose flour, and fine salt together in a large mixing bowl.
Pour the yeast mixture and the olive oil into the bowl with the flour mixture. Stir until the ingredients are combined and a dough has formed.
Lightly flour a work surface, then turn the dough onto the surface. Knead the dough for 6 to 8 minutes, adding flour as needed, until the dough is smooth and elastic.
Coat the inside of a large bowl with olive oil. Roll the dough into a ball, then place it in the bowl. Cover the bowl with a damp tea towel, then set it in a warm location. Let it rise until it doubles in size, which will take 1 to 2 hours, depending on the temperature.
Generously brush the bottom and sides of a rimmed 10 x 15 inch sheet pan with olive oil.
Punch the dough down, then transfer it to the sheet pan. Spread it out into an even layer using your hands. Press your fingertips into the top of the dough to create indentations all over the top surface.
Cover the dough with a damp tea towel and place it back in a warm location. Let it rise again until it has gotten visibly puffier, 30 to 45 minutes, depending on the temperature.
Preheat the oven to 425°F.
Arrange the zucchini slices, bell pepper, cherry tomatoes, red onion, and garlic cloves all over the top surface of the dough, gently pressing the pieces in. Brush the top of the dough and vegetables with 2 tablespoons of olive oil.
Bake the loaf until the vegetables are tender and the bread is puffy and golden brown, about 20 to 24 minutes.
Remove the loaf from the oven and place it on a cooling rack. Sprinkle the top with flaked sea salt and black pepper to taste.
Let the loaf cool for at least 1 hour. Sprinkle the top with fresh basil leaves just before serving. Slice and serve.