Bring a pot of water to a boil. Once boiling, carefully add the eggs. Cook for 7½ minutes, then remove them from the pot and place them in an ice bath to cool down.

In a small bowl, mix the gochujang paste, mayonnaise, sesame oil, and brown sugar (if using) until well combined. Taste and add more gochujang if desired.

Peel the cooled eggs and rinse off any remaining pieces of shell. Place the peeled eggs in a medium bowl and mash them with a fork. Stir in the sauce mixture. Season to taste with salt and freshly ground black pepper.

To serve (optional), toast the sourdough bread and spread it with butter. Divide the kimchi between the toast slices. Spoon the gochujang egg mixture over the toast and top with toasted sesame seeds and chives or green onions.