Let the prepared pizza dough sit for at least 30 minutes and up to 3 hours. Preheat the oven to 500° and place a pizza stone in the bottom of the oven.
Mix the garlic with the lemon juice and mayonnaise in a small bowl. Season with salt to taste.
Heat one tablespoon olive oil in a medium frying pan. Add the onion, cover and cook over medium heat, stirring occasionally, until soft (approximately 8 minutes). Check seasoning.
To assemble, spread the pizza dough on a lightly floured work surface to a 12-inch round. Transfer the round of dough to a pizza pan and brush with olive oil. Sprinkle half the onion, tomatoes and Tonnino Tuna on the pizza and slide it into the hot stone. Bake for 7 minutes until crusty and bubbling. Remove from the oven and sprinkle the pizza with half the aioli.
Cover with half the basil.
Cut into pieces and serve.