Remove Dutch oven from oven and transfer shortribs to a sheet pan.
Strain onion mixture through a fine mesh sieve into a large liquid measuring cup. Discard solids (see notes). Skim grease from top of broth; reserve 2 tablespoons grease then discard remaining grease. Reserve and set aside resulting broth.
Heat 4 teaspoons toasted sesame oil and 2 tablespoons reserved drippings (or butter) in now-empty Dutch oven over medium-high heat. Add 8 ounces mushrooms and cook until golden brown and fried, about 15 minutes, stirring occasionally.
Transfer mushrooms to a bowl and season lightly with kosher salt.
Add remaining 2 teaspoons toasted sesame oil to now-empty pot. Add ⅓ cup minced ginger, 2 tablespoons minced garlic and 2 tablespoons miso paste. Cook, stirring constantly, until fragrant, about 2 minutes.
Stir in reserved broth, 10 cups beef broth, 2 tablespoons fish sauce, and 2 tablespoons soy sauce; bring to a simmer.
Meanwhile, bring a saucepan of water to a boil. Using a slotted spoon or spider, gently lower 4 eggs into water; return to a boil then cover and cook for 6½ minutes. Prepare an ice water bath. Drain eggs then transfer to ice water bath and cool completely.
Fill saucepan with water again and bring to a boil. Add 12 ounces ramen noodles and cook according to package directions. Drain and rinse with cold water.
Peel cooled eggs and cut in half.
Remove meat from short ribs and shred into large pieces. Add to simmering broth.
Divide noodles between 4 bowls. Divide broth between bowls. Top each bowl with a quarter of cooked mushrooms, 2 egg halves, scallions and sesame seeds.
Serve with your favorite hot sauce.