In a large bowl, beat egg, salt, and pepper to blend. Using a food processor or the large holes on a hand grater, coarsely shred zucchini, carrot, and potato. Add shredded vegetables, 1/2 cup basil, and flour to egg mixture; stir just to combine.
Pour 1 tablespoon oil into a 10- to 12-inch nonstick frying pan over medium-high heat; when hot, drop batter in 1/4-cup portions into pan, three or four at a time, and flatten with bottom of measuring cup or the back of a spoon into about 3 1/2-inch rounds. Cook, turning as needed, until browned and crisp on both sides, 6 to 8 minutes total. Transfer pancakes as cooked to an ovenproof platter and keep warm in a 200° oven. Repeat to cook remaining pancakes, adding more oil to pan as necessary.
In a small bowl, mix sour cream with remaining 2 tablespoons basil. Serve pancakes warm, topped with sour cream mixture (see notes).