Using a food processor fitted with a coarse grating disk, grate the potatoes and onion; discard any large pieces. Transfer the mixture to a clean dish towel and squeeze out any excess liquid, then transfer to a large bowl. Stir in the flour, egg, 3/4 tsp. salt, 1/4 tsp. pepper and the baking powder.
Using side of a knife or a mortar and pestle, mash 1 large clove garlic and pinch salt to a paste. In small bowl, whisk together 1/2 cup mayonnaise, 1 tbsp. EVOO, prepared garlic paste and 1 tsp. fresh lemon juice to make aioli; cover and refrigerate.
In a medium skillet, heat 1/4 inch oil over medium-high heat until hot but not smoking. Working in 5 to 6 batches, drop 2 tbsp. scoops of batter into the pan about 2 inches apart. Using a spatula, flatten the batter into disks. Cook, turning once, until browned and crisp, about 5 minutes. (Lower the heat if the latkes darken too quickly.) Drain on paper towels. Serve with aioli.