Preheat oven to 375 degrees F (190 degrees C). Grease a baking dish with 2 teaspoons butter.
Core and cut cauliflower into large florets. Cut broccoli into large florets; peel and quarter broccoli stalks.
Bring a large stockpot of water to a boil; add cauliflower and boil until tender but still firm, 8 to 10 minutes. Transfer cauliflower to strainer using a slotted spoon; run under cold water to stop cooking. Add broccoli stalks to stockpot and boil 3 minutes; add broccoli florets to stockpot and boil until stalks and florets are tender but still firm, about 5 minutes more. Transfer broccoli to strainer and run under cold water to stop cooking. Dry cauliflower and broccoli with paper towels and place in prepared baking dish.
Melt 1/2 cup butter in saucepan over medium heat. Whisk in flour to form a thick paste. Continue whisking constantly until flour begins to brown, about 4 minutes. Add milk and bring to a boil, whisking constantly, until sauce thickens, 2 to 4 minutes more. Remove sauce from heat; whisk in mustard, Worcestershire sauce, and pepper. Whisk in Cheddar cheese until melted and fully incorporated. Pour cheese sauce over cauliflower and broccoli. Rap baking dish on counter a few times to remove air bubbles.
Mix bread crumbs and olive oil together in a small bowl. Top vegetable and sauce mixture with bread crumb mixture. Sprinkle paprika over bread crumb mixture.
Bake in preheated oven until bubbly and golden brown, about 30 minutes.