Working with 1 artichoke at a time, remove tough outer leaves to expose light yellow core. Using knife, trim off top. Using spoon, remove choke. Trim stem end; cut artichoke lengthwise into ¼" think slices. Place in bowl; add half of buttermilk to coat. Place lemon slices in another bowl; coat with remaining buttermilk.
In large bowl, whisk together both flours and 2 tsp. salt. In large Dutch oven over medium-high heat, pour in oil to depth of 2"; heat until it reads 375°F on deep frying thermometer.
Working in batches, drain artichokes well. Coat with flour mixture; shake off excess. Fry artichokes until golden and crispy, 3-5 mins. Transfer to paper towl-lined baking sheet; season with salt. Repeat with lemons, frying 1-3 mins. Arrange artichokes and lemons on warmed platter. Serve with aioli.
