Whisk flour, salt, and black pepper in a bowl. Press chicken thighs into seasoned flour to coat. Heat vegetable oil in a skillet over medium heat; fry coated chicken in the hot oil until golden brown, 5 to 8 minutes per side.
Place potatoes, carrots, and cabbage into a large casserole dish and lay chicken thighs on top of vegetables. Stir beef stock, stout beer, and brown sugar in a bowl. Mix cornstarch and water in a separate small bowl; stir into beer mixture. Pour beer mixture over chicken and vegetables. Cover the casserole dish.
Bake in the preheated oven for 1 hour and 15 minutes; uncover casserole and continue baking until chicken is cooked all the way through and sauce has thickened, 45 more minutes.
