Stir the warm water, yeast, and sugar together in a large bowl. Let it stand for 5 minutes until foamy.
Add the flour, salt, and olive oil. Stir until a shaggy dough forms, then knead on a lightly floured surface for 6 to 8 minutes until smooth and elastic.
Place the dough in an oiled bowl, cover, and let it rise in a warm place for 60 to 90 minutes until doubled in size.
Combine the crushed tomatoes, garlic, and salt in a small bowl. Place a pizza stone or inverted baking sheet in the oven and preheat to 500 F for at least 30 minutes.
Stretch the dough into a 12-inch round on parchment. Spread the tomato sauce over it, leaving a 1-inch border, then add the dairy free mozzarella-style shreds.
Slide the parchment and pizza onto the preheated surface. Bake for 9 to 12 minutes until the crust is browned and the topping is bubbling.
Drizzle with olive oil, scatter the fresh basil over the top, and cool for 5 minutes before slicing.