Heat vegetable oil in a deep skillet over medium-high heat; fry wonton strips until crisp, 2 to 4 minutes. Transfer to a paper towel-lined plate to drain.

Preheat oven to 450 degrees F (230 degrees C). Spread pizza dough onto a baking sheet.

Bake in the preheated oven until slightly brown, about 7 minutes.

Mix cream cheese, 1/4 cup green onions, 2 tablespoons Parmesan cheese, and crab together in a bowl; spread onto the pizza crust. Top with mozzarella cheese, remaining Parmesan cheese, fried wonton strips, and remaining green onions.

Bake in the preheated oven until cheese is browned and melted, about 8 minutes. Drizzle sweet chile sauce over top.