In a medium bowl, mash the 2 cups blackberries. Add to a large ziplock bag, along with the olive oil, 1/2 cup balsamic vinegar, hot sauce, 2 tablespoons white sugar, and steak seasoning. Add the chicken and shake it up so everything gets all mixed together.
Refrigerate for at least 4 hours, or up to 24. Turn every couple hours.
Heat grill to medium heat. If your chicken breasts are very thick, use a rolling pin or meat mallet to pound them down a bit. They will cook more evenly that way.
Prepare the glaze: in a small saucepan combine COLD water and cornstarch with a whisk. Turn the heat to medium. Add 1/2 cup sugar, 1/4 cup balsamic vinegar, and 1 cup blackberries. Mash the berries. Stir over medium heat until the mixture thickens, about 4 minutes. Set over low heat and stir occasionally to keep warm while you cook the chicken.
Grill the chicken for 3-5 minutes, then flip and cook for another 2-4 minutes or so. I highly recommend a meat thermometer.
Serve the hot chicken with the warm glaze spooned over the top. Goes great with grilled corn and mashed potatoes!