Season chicken with salt and pepper. In a skillet, heat the EVOO over medium-high heat. Add the chicken and cook, turning once, for 10 minutes. Remove to a plate, cover to keep warm.
Add the butter to the pan over medium-low heat. Stir in the shallots, garlic and chile; season with salt and pepper. Cook for 3 minutes. Whisk in the flour, paprika and mustard powder. Whisk in the stock and vermouth (or white wine); bring to a bubble. Whisk in the hot sauce, parsley and Worcestershire.
Add the chicken and its juices to the gravy and turn to coat completely. Serve the chicken on the toast points.