Thinly slice the green onion on a bias and set aside for garnish. Bring 2 cups of water to a rolling boil in a pot over high heat. Once boiling, add the ramen noodles and cook for 2 minutes, stirring occasionally to prevent sticking. The noodles should still have a slight firmness since they'll continue to cook in the creamy broth.

Drain the cooked noodles completely and return them to the pot. Pour in the 1.25 cups of milk and add the seasoning packet from the ramen. Stir continuously for about 1-2 minutes until the seasoning dissolves completely into the milk. I find using whole milk like Fairlife gives the sauce a richer, creamier quality than lower-fat alternatives. Add the butter and cheddar slice, stirring until both melt and combine into a smooth, creamy broth.

Turn the heat to medium and let the broth gently bubble and simmer for about 2-3 minutes, stirring occasionally. The mixture will thicken slightly as it heats through. Add the 1/4 cup chopped kimchi and stir to distribute it evenly throughout the noodles. I like to add the kimchi at this point so it softens slightly in the heat but retains some of its spicy, tangy character. Remove from heat immediately after the kimchi is incorporated to prevent the milk from scorching.

Quickly crack the egg directly into the hot noodle mixture and stir vigorously for about 15-20 seconds. The residual heat will cook the egg into creamy ribbons that blend into the cheese broth, adding richness and body. Immediately transfer to a bowl and top with the sliced green onions. Serve right away while the broth is hot and creamy.