In a large bowl, gently mix the beef, pork, minced garlic, 1 tsp. of thyme, 1⁄4 tsp. of paprika, 1 tsp. salt, and 1⁄2 tsp. black pepper. Add the manchego or parmesan, egg, and panko, and gently mix until the meat absorbs the breadcrumbs.
To form the meatballs, set a small bowl of cold water nearby and, occasionally moistening your hands, gently roll 1 ½ -oz. portions of the meat into balls; you should get about 20.
Heat 2 Tbs. of the oil in a large heavy-duty Dutch oven over medium-high heat until shimmering hot. Add half of the meatballs, reduce the heat to medium, and cook, undisturbed, until browned on the bottom, 1 to 2 minutes. Turn and brown the other side, 1 to 2 minutes more. Transfer to a paper-towel-lined plate, and repeat with remaining meatballs.
Pour off all of the fat. Add the remaining 1 Tbs. oil and the sliced garlic to the pot. Return the pan to medium heat and cook until the garlic sizzles, about 1 minute. Add the remaining 1 tsp. thyme, and ⅕ tsp. paprika; cook, stirring, for 15 seconds. Add the chicken broth and crushed tomatoes, and bring to a simmer. Return the meatballs to the pot, reduce the heat to medium low, cover, and simmer until the meatballs are cooked through (160°F), about 25 minutes. Season the broth with salt and pepper to taste.
Meanwhile, bring a medium pot of well-salted water to a boil. Add the pasta and cook, stirring, until tender, about 4 minutes. Drain well, return to the pot, and toss with 1⁄2 cup of the broth.
To serve, mound the pasta in shallow bowls. Top with the meatballs and broth and serve.