Cook bacon in a large skillet over medium heat until crispy. Remove bacon, drain on paper towels, then crumble and set aside. Leave bacon fat in the skillet.
Add ground beef to the skillet and cook in the bacon fat until browned and cooked through, breaking it apart as it cooks; season with half the pepper.
Add shredded cabbage to the beef in the skillet, stirring to combine. Cook until the cabbage softens, about 8 minutes.
Sprinkle goat cheese, Swiss cheese, and crumbled bacon evenly over the cabbage and beef mixture. Reduce heat to low and cover the skillet to allow the cheeses to melt gently.
While cheese melts, whisk eggs with the remaining black pepper until frothy.
Pour the egg mixture over the cheese-melted hot mess in the skillet. Cover and cook over low heat until the eggs are just set and form a creamy sauce on top, about 5 minutes.
Remove from heat and let it rest for 2 minutes before serving to let the flavors meld.