Saute' mushrooms in vegetable oil in a skillet that has been heated to medium. About 5 minutes is needed unless mushroom slices are very thin or very thick. Remove mushrooms and set aside.
Salt and pepper chicken breasts on both sides and cook in the same pan as mushrooms 4-5 minutes per side or until golden brown. Add 1 Tablespoon additional vegetable oil if needed. Remove chicken and set aside.
Melt butter in skillet and add shallot and garlic, stirring constantly for about 1 minute.
Whisk flour into cream and add to skillet. Cook and stir constantly until thickened.
Add Dijon and Gorgonzola, stirring constantly until cheese is melted. Add chicken and mushrooms back to skillet and heat thoroughly before serving.
Notes
Boneless chicken thighs may be used but will not need to be pounded.
An alternate blue cheese may be used.
Butter can be salted or unsalted - adjusting added salt accordingly.
Mushrooms may be considered an optional ingredient.
Additional Gorgonzola crumbles can be used as a garnish before serving.