Place 4 slices bacon in a large Dutch oven. Cook over medium-low heat, flipping once, until crisp, 10 to 12 minutes. Transfer the bacon to a paper towel–lined plate and set aside to cool. Pour off and reserve the rendered fat, leaving 2 to 3 tablespoons in the pot.

Add sliced onion, chopped carrots and sliced cabbage to the bacon fat in the pot; cook over medium-low heat, stirring occasionally, until the onion is translucent and the cabbage has softened slightly, 13 to 15 minutes. Season with ½ teaspoon each salt and pepper. Transfer the vegetables to a large bowl.

Add 2 tablespoons of the reserved bacon fat to the pot (discard the rest). Sprinkle in 2 tablespoons flour; cook, stirring constantly, until golden and nutty, 2 to 3 minutes. Gradually add 1 cup chicken stock, whisking constantly until fully incorporated. Gradually add the remaining 7 cups stock, whisking until the mixture is smooth.

Chop the cooled bacon into 1-inch pieces. Add the chopped bacon and the sautéed vegetables to the pot. Stir in ½ cup bulgur (or kasha). Bring to a boil over medium-high heat; reduce heat to maintain a gentle simmer. Cover and simmer until the cabbage and bulgur are tender and the broth has thickened slightly, about 30 minutes. Add the rinsed black-eyed peas; cook until just heated through, 2 to 3 minutes.

Ladle the soup into bowls and top each serving with 2 tablespoons sauerkraut.