Coat a large frying pan with oil and set over medium-high heat. Lightly sprinkle both sides of chicken with salt, then add to pan. Cook, turning occasionally, until golden and springy when pressed, 12 to 14 min in total. Remove to a plate.
Meanwhile, prepare peaches and shallot. After removing chicken from pan, reduce heat to medium. Add wine, then scrape up and stir in any brown bits from pan bottom. Add peaches and shallot. Stir often, until peaches soften slightly and wine is reduced, 2 to 4 min.
Stir in mustard, then whipping cream. Return chicken and any juices to pan and boil gently, uncovered, turning chicken occasionally until sauce thickens slightly and chicken is hot, 1 to 2 min. If using sour cream, add after the chicken is hot. Taste and add salt if needed. Sprinkle with chives or sprigs of fresh dill. Lovely with rice and green beans.