Heat oven to 375°F. Fit pie dough into a 9-inch pie plate, prick, and blind-bake 10 minutes. Set aside to cool slightly.
Melt 2 tbsp butter in a skillet over medium heat. Cook onion, celery, and bell pepper until tender, 4 to 5 minutes.
Stir in garlic and cook 30 seconds until fragrant.
Add chopped shrimp. Sauté until just pink, 2 to 3 minutes. Transfer mixture to a bowl.
Sprinkle flour into the empty skillet. Cook 1 minute, stirring, to make a light roux.
Whisk in milk gradually until smooth. Simmer until slightly thickened, 2 to 3 minutes.
Remove from heat. Whisk in mayonnaise, lemon juice, Old Bay, hot sauce, salt, and black pepper.
Fold in cheddar and parsley until the cheese starts to melt. Combine sauce with the shrimp and vegetables.
Spoon filling into the par-baked crust. Mix cracker crumbs with melted butter and paprika; scatter evenly over the top.
Bake until bubbly and golden, 18 to 22 minutes. Let rest 10 minutes before slicing.