Inspect squash blossoms for dirt, or bugs. Gently rinse with water and pat dry with paper towels. Remove the pistil from inside each flower.
Saute blossoms with the garlic in olive oil over medium high heat until softened and lightly browned. Remove from the skillet and coarsely chop. Transfer to a bowl. Add basil, Parmesan, egg, and milk and gently beat with a fork. In a separate bowl, whisk together the flour, baking powder, and salt. Pour the wet ingredients into the bowl, and stir to combine. The mixture should be slightly thicker than pancake batter. If needed, add in a bit more milk.
Pour enough olive oil into a skillet to coat the bottom by about a 1/4"; heat over medium high heat. Scoop the batter by the heaping tablespoonful, and drop into the hot oil. Use the back of a spoon to flatten each fritter slightly. Cook until golden brown, about 3 minutes. Flip the fritter over, and brown the other side. Transfer to a paper towel lined plate. Sprinkle lightly with some coarse salt and chopped fresh basil. Serve while hot.