Heat the olive oil in a medium saucepan over medium heat. Add the shallots and cook for about 3 minutes or until just tender, stirring occasionally. Add the blackberries, thyme and chicken stock. Bring to a boil. Reduce heat and simmer for about 20 minutes or until the blackberries have broken down. Press the blackberry mixture through a fine sieve over a bowl, discarding solids. Return the blackberry sauce to the pan and stir in the vinegar and honey. Bring to a boil. Cook until reduced to about 3/4 cup. Remove from the heat. Whisk in the chilled butter and adjust seasonings to taste with salt and freshly ground black pepper. Set aside.
Meanwhile, season the chicken breasts on both sides with salt and freshly ground black pepper. Spray both sides with olive oil. Place the chicken on the grill and grill for about 4-5 minutes per side or until cooked through. Transfer chicken to a platter and tent with foil; let rest for 5 minutes. Serve chicken with blackberry sauce drizzle over the top. Garnish with fresh thyme sprigs if desired.