BATTER: In a mixing bowl, whisk together flour, cornstarch, baking powder, season salt, and black pepper; whisk in the egg and ice-cold water, a little at a time, until batter is smooth and has the consistency of pancake batter. Refrigerate the batter for 15 to 20 minutes.
While the batter is chilling, prepare the squash blossoms. Carefully separate the flower petals without breaking them and remove the stamens or pistil in the center.
FILLING: In a small bowl, combine goat cheese, egg yolk, Gruyere cheese, black pepper, and cayenne pepper together; stir until smooth. Spoon filling into a heavy, re-sealable 1-quart plastic bag, squeeze out the air, and seal the bag. Cut a small corner off the bag.
Gently insert the cut corner of the bag all the way to the bottom of the open end of a blossom and pipe about 1 tablespoon of filling inside. Pick up petals and drape them up over the filling, covering filling completely. Fold any excess petals over the top of the filled blossom to keep them out of the way. Refrigerate filled squash blossoms until cheese is set and firm, at least 30 minutes.
Pour vegetable oil into a heavy cast iron skillet, about1-inch deep and place over medium heat. Heat oil until a thermometer placed into the oil reads 350 degrees. A drop of batter carefully dripped into the oil should sizzle immediately.
Remove squash blossoms from refrigerator and dust lightly with all-purpose flour on all sides. Shake off excess flour. Dip 6 blossoms into batter. Let excess batter drip off.
Carefully place each batter-covered blossom into the hot oil and fry until golden crisp on both sides, about 1 minute on the first side and 30 seconds to 1 minute on the remaining sides.
Remove and drain on paper towels, then season with salt and pepper. Repeat with remaining blossoms. Let cool slightly before serving.
Cook’s Note: The blossoms that come attached to a small baby squash are female blossoms; the ones without a squash are male blossoms. Both are delicious. Pick them in the morning or buy the freshest flowers you can find and cook them the day picked or purchased; the longer you store the blossoms, the more the petals will stick together, making it harder to stuff them.