Peel and mince the shallots and reserve.
Clean and pick over the trumpet mushrooms (if using). If using white or brown mushrooms, quarter these to cook along with the shallots. Reserve
Core a medium size tomato. Cut an X into the bottom. (The X will make it easier and serve as a starting point to peel the tomato.) Submerge in boiling hot water for 10 seconds. Carefully remove from the hot water and shock in ice water.Use a paring knife to peel back the skin starting at the X and remove. Once peeled, cut the tomato through the horizontal (see image) and carefully squeeze out the seeds, (you may need to use your finger to get all the seeds sacs out). Keep the tomato mostly intact. Slice into thin strips and dice into cubes. Reserve
Wash and clean the tarragon, while pulling the leaves off the woody stem. Chop into a fine dice. Wash the chives and chop into thin slices. Do the same for the parsley. Reserve.
On a cutting board, trim fat or cartilage off the chicken breast. (Clean and sanitize your cutting board.)
Place the boneless skinless chicken breast (one at a time) in a gallon sized Ziploc or between two pieces of plastic wrap. Use a meat mallet or tenderizer to pound out to a uniform thickness while keeping the basic shape of the breast.
In a medium sized kitchen bowl, add the flour and season with salt and pepper. Stir to incorporate.
Remove the chicken breast from the plastic and dredge the chicken breasts in the seasoned flour. Reserve.
Heat a large sauté pan over medium to medium-high heat with the butter and olive oil. Test heat by dropping a pinch of dredging flour into the butter/oil mixture. It is ready when the flour starts to sizzle. Carefully add the chicken breasts and cook until one side is golden brown. If the oil is too hot (dark bits form in the pan), remove the saute pan from the heat and lower the temperature to continue.
With a set of tongs, carefully turn the chicken breast and cook the second side.
Once the second side is golden brown, remove the chicken to another bowl or plate and reserve.
Add the peeled and minced shallots to the pan that sauteéd the chicken and cook for 2 minutes. Add the mushrooms and a pinch of white pepper and salt. Cook until the shallots and mushrooms are tender.
Deglaze the pan with Cognac and sherry vinegar. The alcohol may ignite while cooking in this step, but it will cook off very rapidly and reduce to almost nothing. When this happens, add the chicken stock and stir.
Once the chicken stock has reduced to half, add the cream and stir into the shallot/mushroom mixture. Add the sauteed chicken breast back into the sauce. Turn the chicken over to thoroughly coat the chicken breast. Continue cooking for 2 minutes and add the tarragon and chives. Cook for 1 minute more.
The chicken should be fully cooked by now. If you are unsure, test with a meat thermometer. The internal temperature should be 165°F or 75°C. Add half of the chopped tomato. If the sauce is too thick you can add a bit more chicken stock or cream to thin out. If the sauce is too thin, lower the heat and let it cook out till thickened or add some a small amount of cornstarch mixed with water.
Serve the chicken with the buttery rice or your favorite side dish. Top with a light sprinkling of reserved dice tomato and parsley.