Using tweezers or a small, sharp knife, carefully remove and discard pistil from each zucchini blossom.
Trim stems to about 1 1/2 inches.
Pat mozzarella and prosciutto with paper towels to remove excess moisture.
Carefully stuff zucchini blossoms with mozzarella, prosciutto, and basil.
Lightly twist flower at the top to help close.
Pour olive oil to a depth of 1/4 inch in a large cast-iron skillet; heat over medium-high to 350°F.
Whisk together egg and 1/4 tsp salt in a medium bowl.
Stir together flour and baking powder in a small bowl.
Gradually whisk half of flour mixture into egg mixture.
Add remaining flour mixture alternately with seltzer water, whisking until batter is smooth.
Dip 4 stuffed zucchini blossoms into batter, coating completely and allowing excess to drip off.
Fry in hot oil, turning occasionally, until golden brown, 4 to 5 minutes.
Transfer to a paper towel–lined baking sheet, and sprinkle with 1/8 tsp salt.
Repeat with remaining zucchini blossoms, batter, and salt.
Serve immediately.