Make the batter: Stir, using a whisk, 1 cup flour, salt, and pepper in a medium bowl. Whisk the egg and the beer together in a small bowl. Slowly pour the egg mixture into the flour mixture, stirring constantly with a fork, until almost combined. The batter will be slightly lumpy. Let stand for 30 minutes.
Fill and fry the blossoms: Heat oil in a 4-quart Dutch oven fitted with a deep-fat thermometer over medium-high heat to 375°F Gently open the petals of the blossoms using a paring knife and carefully remove the pistils without tearing the petals. Combine the remaining flour and cayenne pepper in a medium bowl, add the cheese, and toss until the cubes are coated. Place one cube inside each blossom and twist the ends to close. Dip blossoms, holding them at the petal end, one at a time into batter. Wipe excess batter from blossom against the side of the bowl.
Carefully ease 3 blossoms into hot oil and fry, turning once with a slotted spoon until golden brown — about 4 minutes. Remove with a slotted spoon and drain on a wire rack. Repeat with remaining blossoms. Serve immediately with lemon wedges.