Preheat a 6-quart Dutch oven over medium heat. Add avocado oil to the warm pan and heat until it shimmers, about 1 minute. Add sausage pieces, cut-side down, and heat until browned, 1 to 2 minutes. Turn and brown on the other side, about 1 to 2 minutes more. Transfer sausage to a plate and reserve juices in the pan.

Add carrots, potatoes, and onion to the Dutch oven and stir to coat with pan juices. Add chicken broth and black pepper, stirring to combine. Return browned sausage to the pan on top of the vegetables, and place cabbage wedges on top of kielbasa. Bring broth to a boil, about 5 minutes, then cover the Dutch oven and reduce heat to a simmer. Cook until vegetables are fork tender, about 20 minutes.

Use a slotted spoon to remove all vegetables and sausage pieces and set aside in a bowl. At this point, cabbage may be cut into bite-sized pieces.

Place flour in a small bowl and add enough water to make a thin paste, stirring until all lumps dissolve. Whisk liquid in the Dutch oven constantly while gradually adding flour mixture until liquid boils again, about 5 minutes. Return vegetables and sausage to the Dutch oven and adjust seasoning, if necessary. Serve in individual bowls.