Heat oil in a wide pot over high heat until it reaches 350° on a deep-fry thermometer. Reduce heat to low. Whisk flour, beer, salt, garlic powder, and cayenne together in a medium bowl. Working in 2 batches and using tongs, drop vegetables into batter and then into oil. Fry until golden brown, about 4 minutes. Drain on paper towels.
Mix mayonnaise and mustard together and serve on the side.
Note: Nutritional analysis is per serving.