Melt the butter in a small saucepan over medium heat. Stir in the flour until smooth, and cook for 10 minutes, stirring frequently to make a roux. Set aside when done.
Meanwhile, heat the vegetable oil in a large pot over high heat. Stir in the chicken breast, and cook until it begins to brown a bit on the edges, about 5 minutes.
Stir in the vegetables, and cook until the onions are tender, about 7 minutes or so. Stir in the spices and chicken stock next. Bring to a simmer, then stir in the flour and butter mixture. Reduce the heat, and simmer 30 to 40 minutes until the potatoes and vegetables are tender.
Once the vegetables are tender, stir in the Cheddar cheese until melted, then the milk. Reheat and serve.