Preheat oven to 275 degrees. Melt 1 tbsp butter in saucepan add 1 tbsp. of oil. Saute the shallots until translucent about 2 min. Add minced garlic and saute 30 seconds. Remove from pan and set aside.
Mix flour with 1 tsp. salt, the pepper and garlic powder. Dredge the chicken breast in flour mixture shaking off excess. Combine the remaining butter and oil in same skillet and cook chicken 4 to 5 min. each side. While chicken is cooking get another saucepan and throw in the shredded cabbage. Season with salt. Add 1/2 chicken stock and cook until cabbage is wilted. Taste and adjust seasoning. Remove from heat and keep warm. Transfer chicken to a baking sheet and put in oven to keep warm. Meanwhile bring a large dutch oven filled with salted water to a boil and cook linguini.
Deglaze the saute pan with the wine, scraping up the bits from the bottom. Cook and reduce about 3 min. Add the mustards and the other 1/2 cup chicken stock and stir until smooth. Stir in the cream. Bring sauce to a simmer over low heat.
Add the chicken back to the sauce. Top with the cabbage and shallot, garlic mixture.Top with cheese. Set the oven on broil and broil just until cheese is melted. Serve over Linguini.