Cut chicken breasts in half lengthwise; pat them dry.
Add the oil and 1 tablespoon of the butter in a large skillet over medium heat. Dredge the chicken in the flour and brown in the hot skillet. Cook each side until it’s golden brown and cooked through, about 10 minutes total time. The chicken should be 165 F/74C at the thickest part. Remove the chicken to a plate and cover with foil.
Add the remaining butter to the same skillet, let it melt then add the sliced mushrooms. When the mushrooms start to release water, add the garlic and Italian seasoning. Continue cooking until the water has evaporated. Remove the mushrooms from the pan—you can put them on the same plate as the chicken if you wish.
Add the chicken stock, Dijon mustard and the lemon juice to the skillet. Let this reduce by 50%. Add the cream, stir well then add the chicken and mushrooms back to the pan. Cook for about 5 minutes to heat through letting the cream reduce a bit— to the consistency of a thin gravy.
If you're using thinner, smaller chicken breasts, you'll need 6.