Heat butter and olive oil in a large skillet over medium-high heat.
Add about ½ the chopped shallots to the pan and stir to coat. Let cook 1-2 minutes, then add the chicken breasts to the pan. Sear until golden brown on one side, about 5-6 minutes. Then flip and sear the other side, about 2-3 minutes.
Add 2 tbsp of water to the pan, cover, and reduce heat to medium-low and cook for about 10 minutes, until chicken is cooked through.
Once chicken is cooked through, remove from pan onto a plate; keep warm.
Add the remaining shallots to the drippings in the pan. Stir to coat, increase heat to medium, and let shallots cook about 2-3 minutes. Then reduce heat to low and add the chopped thyme. Sautee another 1-2 minutes, stirring constantly, until thyme is fragrant.
Add the jam mixture to the pan and stir to combine with shallots and thyme. Bring to a simmer on low heat, then simmer gently for 3-4 minutes, stirring often.
Add the cooked chicken back into the pan, along with any juices on the plate. Stir sauce until combined, and turn chicken breasts over in sauce to coat. Let simmer another 3-5 minutes until chicken is warmed through.
Serve immediately with generous helpings of sauce.