Heat vegetable oil in a deep-fryer or large saucepan to 350 degrees F (175 degrees C). Dry sliced bell pepper, onion, and zucchini on a paper towel.
Whisk flour, baking powder, and salt together in a large bowl; add beer and sparkling water. Stir until batter just combined.
Working in batches, dip bell pepper, onion, and zucchini slices into batter, draining excess; fry in preheated oil until golden brown, 2 to 4 minutes per side. Transfer vegetables to a plate lined with paper towels.
Dip chicken breasts into remaining batter and fry in oil until golden brown, about 6 to 8 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Drain excess oil on paper towels.