Heat the oil and butter in a large skillet over medium heat. Add the cabbage, the pepper, and 1/2 tsp of the salt and sauté until soft and golden, 5 to 6 minutes. Add the peas and cook another minute. Remove from skillet to a plate, cover with foil, and set aside.
Add the broth, lemon juice, and remaining salt to the skillet and bring to a simmer. Place the cod in the pan and poach, covered, until opaque, about 4 minutes total, flipping carefully once. Transfer the cabbage and cod to plates.
Boil the poaching liquid over high heat until slightly thickened, about 2 minutes. Spoon over the cabbage and cod before serving.