Press blackberries through a sieve with the back of a spoon to remove the seeds. Reserve resulting puree - about 1/2 cup. Remove all visible fat from chicken breasts, rinse well and pat dry with paper towels. Sprinkle both sides with pepper.
Heat 1 tablespoon of the olive oil until shimmering in a large, nonstick skillet. Saute chicken breasts until lightly browned on each side and mostly done - about 3-4 minutes per side. Remove from pan.
Add remaining 1 tablespoon olive oil to pan. Stir in shallots and mushrooms and saute 1 minute. Add 1/4 cup of the chicken broth and continue cooking about another minute - or until shallots are softened but not brown.
Combine cornstarch with remaining cup of chicken broth and add to vegetables. Stir in reserved blackberry puree and return chicken breasts to pan. Cook until sauce is thickened and bubbly and chicken is done so the juices run clear.