In a large pot, heat olive oil over medium heat. Add the onion and garlic, cook while stirring occasionally until starting to brown, 4 to 5 minutes.
Add the beef and break up the meat into smaller clumps with the edge of a spatula. Continue to cook until the meat is no longer pink, 4 to 6 minutes. Stir in the tomato paste and cook for another 1 to 2 minutes.
Add the water, apple cider vinegar, and soy sauce and bring to a simmer. Add salt, cinnamon, black pepper, harissa, cumin, cloves, sugar, and bay leaves. Stir to blend.
Reduce the heat to medium low to maintain a gentle simmer. Cover with lid but leave a gap for steam and allow the chili to simmer until thickened and reduced by about half, stirring occasionally, 60 to 90 minutes. Allow the chili to cook for a few minutes before serving.
Bring a large pot of water to a boil and add the udon to warm through (frozen udon is already cooked). Strain the water and divide the noodles between bowls.
Top the noodles with chili and add an egg yolk to the center. Sprinkle on cheese and green onions. Toss the noodles to mix the cheese and egg yolk with the chili.