Prepare the hush puppies: Heat oil in a large stockpot over medium-high to 350°F.
Stir together cornmeal, flour, salt, baking powder, and baking soda in a medium bowl. Stir together buttermilk, water, and eggs in a large bowl. Add cornmeal mixture to buttermilk mixture, and stir until just smooth. Fold in pimientos, both cheeses, prosciutto, and scallions.
Carefully drop half of batter by 1/4-cupfuls into hot oil. Cook until golden brown, rotating occasionally, about 2 to 3 minutes. Drain on paper towels. Repeat with remaining batter.
Prepare the tartar sauce: Stir together all tartar sauce ingredients; serve with hush puppies.