Preheat a skillet over medium-high heat with butter and oil. Also, preheat oven to 350 degrees F
Meanwhile, season chicken with kosher salt (no pepper until the end, it burns) and dust lightly with flour (Pro-tip: Wondra Quick Mixing Flour gives the best light, crispy crust on chicken and fish)
Once the oil/butter combo looks like asphalt on a hot summer day, add the chicken, skin side down (the first surface contacting the pan always browns the best, so always do presentation side first)
Sear on skin side until golden brown. Adjust the pan hotter if you don’t hear sizzling, colder if you see smoke.
Flip chicken over and sear for the same amount of time on second side.
Once both sides are browned, remove the chicken from the pan using tongs or a slotted spoon. Transfer to another oven-safe pan or sizzler platter, sheet pan, etc. Cook until internal temperature reaches about 165 degrees F (next to bone).
Meanwhile, pour off about 1/2 butter/oil combo, turn heat to low and sweat shallots until they are softened (about 2 minutes).
Add vinegar and turn heat back to medium-high. Reduce until almost dry (au sec).
Add chicken stock. Reduce by approx. 3/4, adding tomatoes once you get to about 1/2 reduced. You’ll be left with about 1/2 cup of sauce. Pro tip: For extra sheen and body for the sauce, add about a tablespoon of cold butter to the pan, shaking pan assertively from front to back (whisking adds air and doesn’t give the same sheen).
Toss the cooked chicken in the sauce and serve with mashed potatoes, rice, or just vegetables.