Use a meat mallet or a rolling pin to pound chicken breasts flat to 1/2-in thickness. Pat chicken breasts dry with paper towels and season on both sides with salt and pepper.
Melt 1 tablespoon butter and olive oil in a 9-inch cast iron skillet or stainless steel skillet over medium-high heat. Add chicken to pan and sear 2-3 minutes per side until golden brown. Transfer chicken to a plate and set aside.
Add cherries and shallot to pan and saute in pan drippings over medium-high heat 3-4 minutes until just tender and juicy. Transfer cherries to a plate and set aside.
Melt remaining 1 tablespoon butter over medium-high heat. Add garlic and saute 1-2 minutes until fragrant. Add wine, chicken stock, and lemon juice and bring to a simmer over medium-high heat. Reduce heat to medium-low and allow sauce to simmer 5-10 minutes until slightly reduced.
Nestle chicken and cherries back into sauce and allow to simmer over medium-low heat an additional 5-10 minutes until chicken registers 165F.
Season chicken with additional salt and pepper to taste. Break burrata open over chicken and cherries and garnish top with fresh oregano. Serve chicken immediately and enjoy!