Cook Bacon: Bake at 400°F for 15–20 minutes until crispy. Chop and reserve 2-3 Tbsp bacon drippings.
Toast Bread: Reduce oven to 200°F. Toast sourdough slices for 45–60 minutes, flipping halfway.
Blanch Cabbage: Boil cabbage for 1 minute, then drain and set aside.
Sauté Veggies: In a large pot, heat bacon drippings or oil. Sauté onions and carrots for 6–8 minutes until softened.
Simmer Soup: Add cabbage, bacon, broth, caraway seeds, vinegar, salt, and pepper. Simmer 15 minutes.
Make Cheesy Toast: Top bread with Irish cheddar and broil until melted and bubbly.
Serve: Ladle soup into bowls and top with cheesy toast or serve on the side.