Fill a saucepan with water and bring it to a gentle boil. Reduce the heat to a steady simmer.
Carefully crack each egg into the simmering water and poach for 4–5 minutes until the whites are set and the yolks remain soft. Remove with a slotted spoon and set aside.
Empty the pot and add the chicken broth. Bring it to a boil over medium-high heat.
Stir in the sliced kimchi with its juice, gochujang, and gochugaru until fully combined.
Add the instant ramen noodles to the broth.
When the noodles are about halfway cooked, stir in the mushrooms and sesame oil.
Continue cooking until the noodles are tender and the mushrooms are softened.
Taste the broth and season with salt if needed.
Divide the Kimchi Ramen between two serving bowls.
Top each bowl with a poached egg and chopped scallions.
Serve immediately while hot.