Pulse shallot and garlic clove in food processor until minced. Add salt, pepper, and red pepper and pulse once more.
Add packed basil leaves, vinegar, and about 1 tbsp (eyeball it) olive oil to the food processor and begin blending. Add remaining oil in a small stream with the motor running. Process until basil vinaigrette is a smooth consistency, about 1-2 minutes.
Roast beets
Preheat oven to 400 degrees. Place diced beets on a parchment-lined roasting pan and coat with olive oil, salt and pepper. Roast for 40 minutes, or until browned.
Prepare pizza
Preheat oven to 500 degrees with roasting pan or pizza stone inside.
Stretch portions of dough to 8-10 inches wide on parchment paper (or cornmeal if using pizza stone and peel).
Spread 1-2 tbsp basil vinaigrette on dough with the back of a spoon. Top evenly with beets, mozzarella, and goat cheese.
Slide pizza on parchment paper onto hot roasting pan (be careful!) and bake until cheese is bubbly, about 8-10 minutes. Finish with a few basil leaves.