Place the pan over medium-high heat for 2 minutes, then add the caraway seeds and toast until fragrant, 1 minute.
Add half of the butter, the onion, apples, and 1 teaspoon of the salt. Cook for 4 minutes, until softened and just beginning to brown.
Stir in the cabbage, the remaining salt, black pepper, and dulse flakes. Pour in the chicken stock, cover with the lid, and cook for 8 minutes, stirring halfway through.
Uncover the pan and add the lemon juice and mustard. Cook for about 2 minutes then kill the heat and stir in the remaining butter.
Serve with a few more grinds of black pepper, if desired.