Preheat oven to 425°F. Combine ground beef, egg, cheese, bread crumbs, onion, garlic, 1/3 cup of the ketchup, and 1/2 teaspoon of the salt in a bowl; stir gently to combine. Shape into 4 loaves (each 4 inches long, 1 1/2 inches wide, and 2 inches tall). Place loaves on a rimmed baking sheet. Bake in preheated oven until a thermometer inserted in thickest portion of loaves registers 125°F, 8 to 10 minutes.
Toss together cauliflower, oil, paprika, and remaining 1/4 teaspoon salt in a bowl. Remove baking sheet from oven; arrange cauliflower on unused parts of baking sheet. Bake at 425°F, until loaves reach 160°F and cauliflower is tender and browned, 15 to 20 minutes.