Bring a pot of generously salted water to boil. Meanwhile, core and quarter tomatoes. Add to blender along with garlic, sugar, and about half the basil, and blend just until coarsely puréed. Add 1 tbsp. oil if needed to loosen it up.
Add remaining basil, 1/2 cup oil, the salt, pepper, chile flakes, and 1 1/2 cups cheese to blender and pulse to coarsely purée the basil.
Grind almonds in a clean coffee or spice grinder until they're about the texture of coarse cornmeal.
Boil pasta until just tender, 2 to 3 minutes for fresh and about 15 for dried. Meanwhile, pour half the sauce into a medium frying pan and bring to a simmer over low heat. Whisk almonds into sauce and let simmer gently until creamy-looking, about 2 minutes. If it doesn't look creamy, add 1 to 2 tbsp. water. Remove from heat.
Pour remaining sauce into frying pan (swirl a bit of water into sauce clinging to sides of blender and add that too).
Drain pasta, reserving 1/2 cup pasta water. Put pasta back in pot and stir sauce into pasta. Reheat over low heat, stirring occasionally and adding pasta water if it looks too thick, until pasta absorbs sauce slightly, about 4 minutes. Serve immediately, sprinkled with more cheese and a few torn basil leaves.
Make ahead: Sauce, up to 1 week, chilled; up to 4 months, frozen.