For the dough: Combine the bread flour, whole wheat flour, sugar, yeast and salt in the bowl of a stand mixer. Mix with the dough hook attachment to combine. Add the water and olive oil and mix on low speed just until the dough comes together. Increase the speed to medium-high and mix, adding just enough additional flour so that the dough no longer sticks to the bowl, until the dough is smooth and slightly sticky, 7 to 10 minutes. (Alternatively, mix by hand and knead on a lightly floured surface.) Place the dough in an oiled bowl and turn it once or twice to coat it in the oil. Cover the bowl with plastic wrap and let rise at room temperature until it has doubled in size, 1 to 2 hours.
For the garlic oil: Meanwhile, combine the garlic cloves, rosemary sprigs, thyme sprigs, bay leaf and oil in a small saucepan. Bring to a simmer over the lowest heat and cook until the cloves are light golden and completely tender, about 30 minutes. Let cool. Remove the herbs and stir in the red pepper flakes.
For the pizza: Preheat the oven to 450 degrees F.
Punch the dough down. Brush a half-sheet pan with the olive oil (not the garlic oil) and pat out the dough to the edge. (This might seem like a lot of olive oil, but it'll make it good.) Set aside to rise, uncovered, for another 20 minutes or so.
Combine the tomatoes, eggplant, bell pepper, squash and zucchini in a large colander. Sprinkle with 1 teaspoon salt and toss well. Let drain in the sink while the dough rises.
Brush the dough with 2 tablespoons garlic oil and spread the marinara over the top in a very thin layer. Sprinkle with the chopped rosemary and thyme (reserved from making the garlic oil). Fish the garlic cloves from the oil and smooth and scatter them over the top. Pat the vegetables dry, then alternate the slices of the tomatoes, eggplant, bell pepper, yellow squash and zucchini over the sauce, shingling them in a pretty pattern. Tear the burrata in small shreds over the top. Drizzle with another tablespoon or two of the garlic oil. Bake the pizza on the bottom rack of the oven until the bottom crust is crisp and the top is golden, about 25 minutes.
Scatter the basil and arugula over the top. Drizzle with 2 tablespoons of the garlic oil and sprinkle with the Parmesan and flaky salt.