Season chicken with salt and pepper. In a large skillet, heat 1 tablespoon oil over medium. Add chicken and cook until opaque throughout, about 5 minutes per side. Transfer to a plate. Set aside.
Make chutney: To skillet, add remaining 1 tablespoon oil, onion, and jalapeno; cook until softened, about 3 minutes. Increase heat to medium-high; add plums, sugar, vinegar, curry powder, ginger, and 1/4 cup water. Bring to a boil; reduce to a simmer, and cook until plums are softened and liquid is slightly syrupy, about 8 minutes.
Add chicken and any juices that have accumulated on plate; simmer until heated through, about 4 minutes. Season with salt and pepper to taste. Serve chicken with chutney spooned on top.