Brown the sausage first and cook the cabbage in its fat. That rendered smoky fat is where most of the depth comes from. Clean oil will not give you the same pan.
Undercook the pasta. Pull the łazanki a little short of al dente. They keep softening in the pan with the cabbage, and squares that went in soft turn to mush in the tossing.
Taste the sauerkraut before it goes in. They vary enormously. If yours is fiercely sour, rinse and squeeze it; if it is mild, keep a splash of the brine back and add it at the end.
Dried mushrooms turn it into the Christmas version. Soak them, use the soaking water to cook the sauerkraut, and leave the last spoonful behind because the grit settles.