Sprinkle the shrimp in a bowl with the oregano, 1 tablespoon olive oil, 1/2 teaspoon salt and a pinch of pepper.
Heat a skillet over medium-high heat, then add the butter and let melt. Add the flour and cook, 1 minute. Add the garlic and continue to cook, whisking, 1 minute more. Whisk in the milk and cook until very thick, 2 to 3 minutes. Stir in 1/4 cup Parmesan and remove from the heat. Season with a pinch of salt and pepper.
Preheat the broiler to high.
Roll out the pizza dough on a lightly-floured work surface into two 10-inch ovals. Brush the grill pan with the remaining tablespoon olive oil. Grill the dough on both sides until there are grill marks, 2 to 3 minutes per side. Remove the crusts to a sheet tray. Add the shrimp to the grill pan and grill on both sides until fully cooked, pink and opaque, about 4 minutes total. Spread the white sauce on the dough in an even layer. Scatter the shrimp over the sauce. Sprinkle with the mozzarella and the remaining 1/4 cup Parmesan. Place under the broiler until the cheese is melted and bubbly, keeping a close eye so it doesn't burn, 2 to 7 minutes. Sprinkle the red pepper flakes and parsley on top before serving.