Preheat oven to 400F.
Heat a large skillet over medium-high heat with 1 turn of the pan of EVOO. Cook kielbasa until brown and crispy, about 3-4 minutes. Remove the sausage from the pan with a slotted spoon and set aside.
Add the hash brown potatoes into the same pan with the sausage drippings and cook until brown and crispy, about 4-5 minutes.
While the hash browns are crisping, place a medium-size skillet over medium-high heat with one turn of the pan of EVOO. Add the onions, garlic, salt and freshly ground black pepper and cook until tender, 5-6 minutes.
Sprinkle the flour over the ingredients in the pan and cook for about 30 seconds. Whisk in the chicken stock, milk and mustard. Bring the gravy up to a bubble, then reduce the heat and add the sauerkraut. Cook until the gravy has thickened and the sauerkraut is heated through, 3-4 minutes.
Stir the reserved kielbasa into the gravy and sauerkraut, and transfer the mixture to a 9x13 casserole dish. Top with the hash brown potatoes and a sprinkling of shredded cheese. Dot the top of the casserole with the butter then transfer to oven and cook until the cheese has melted.