1. Melt butter in a medium saucepan over medium heat. Add shallots, and cook until softened, 3 to 4 minutes. Add flour, and cook, whisking constantly, for 1 minute. Add milk in a slow, steady stream, whisking constantly, until mixture comes to a boil. Reduce heat to low, and cook, whisking often, until mixture has thickened, about 3 minutes.

2. Add crabmeat, Parmesan, minced parsley, lemon zest, and a pinch of cayenne. Remove from heat, and stir until combined. Season with salt and pepper. Spread mixture on a rimmed baking sheet, and let cool completely. (Cooled mixture can be covered and refrigerated for up to 1 day.)

3. Set a wire rack over a rimmed baking sheet. Whisk eggs in a shallow dish. Place breadcrumbs in another shallow dish. Shape cooled crabmeat mixture into 1-inch balls. Working with 1 ball at a time, coat in beaten egg, then in breadcrumbs. Transfer to rack. Repeat with remaining balls. Let stand, uncovered, at room temperature for 30 minutes.

4. Heat 4 inches of oil in a large, heavy pot until it reaches 375 degrees. Working in batches, fry crab balls, turning once, until golden brown, 1 1/2 to 2 minutes. (Adjust heat as necessary to keep oil at a steady temperature.) Using a wire-mesh skimmer, transfer crab puffs to paper towels to drain, and immediately season with salt.

5. Reduce heat until oil reaches 350 degrees. Fry capers for 1 minute, and using a wire-mesh skimmer, transfer to paper towels to drain. Fry parsley leaves for 1 minute, and using a wire-mesh skimmer, transfer to paper towels to drain. (The capers and parsley may cause the hot oil to spatter when added to the pot.) Sprinkle puffs with capers and parsley. Serve warm.