Combine flour, kosher salt, and yeast in a food processor. Pour in water and olive oil, with the food processor running, until the mixture forms a sticky ball of dough. Add more water if needed.
Turn dough onto a floured surface and knead until it forms a smooth, round ball, about 10 seconds. Place dough in a greased bowl. Cover with plastic wrap and let rise in a warm, draft-free area until it doubles in size, 1 to 2 hours.
Heat 1 teaspoon olive oil and balsamic vinegar in a skillet over medium heat; cook and stir onion and garlic until softened and slightly browned, 5 to 8 minutes. Add apples; cook and stir until slightly softened, 2 to 3 minutes.
Toss radicchio with remaining 1 teaspoon olive oil, salt, and pepper together in a large bowl.
Preheat oven to 450 degrees F (230 degrees C).
Spread dough onto a pizza stone or greased baking sheet.
Bake in the preheated oven until pale, 5 to 7 minutes. Spread a thin layer of onion mixture onto crust and top with radicchio. Bake in the oven for 5 minutes more. Turn on the oven's broiler and broil until edges begin to brown, 1 to 2 minutes.
Remove flatbread from oven and garnish with parsley. Top with a drizzle of honey.
