Combine beef and tomato paste in a large pot. Rinse out tomato paste can with a couple tablespoons of water and add to the pot with garlic, onion, chili powder, cumin, cinnamon, cayenne, allspice, cloves, black pepper, bay leaf, sugar, and salt. Add apple cider vinegar, Worcestershire, and cold water. Mash and stir with a potato masher or a large whisk to break up the meat into fine pieces.

Place the pot over medium-high heat; bring to a boil, stirring occasionally with a wooden spoon. Reduce heat to medium-low and cook until flavors come together and chili reaches your desired thickness, 1 to 1 1/2 hours.

Meanwhile, bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally, until very tender, about 12 minutes. Drain.

Taste chili and adjust seasoning. Remove and discard bay leaf. Ladle over spaghetti into serving bowls. Top with onion and Cheddar cheese. Cut and scoop the noodles up with a fork like you're eating a casserole.