Heat 2 teaspoons olive oil in a medium skillet. Sprinkle the chicken breasts with salt and pepper and cook them through, 3 to 4 minutes per side. Set aside and keep warm. In the same skillet, cook the chicken sausage, turning frequently, until they are cooked through, 10 to 12 minutes. Set aside and keep warm.
Add the shallots to the skillet and cook until softened. Add the berries and cook until they are thawed and have released all their juices. Strain out the seeds and shallots and return the juice to the skillet. Add the balsamic vinegar and reduce the liquid until it has thickened to the consistency of syrup, about 5 minutes. Set aside and keep warm.
Drain the artichokes and pat them dry. In a clean skillet, add 1 tablespoon olive oil over high heat. Add the artichokes and cook until they are browned and crispy, 3 to 4 minutes per side. Drain on paper towels.
Wipe out the skillet, put it on medium heat, and add the remaining 2 teaspoons olive oil. Toss in the zucchini strips and season them with salt and pepper. Toss gently and cook them until they are just warmed through and slightly softened, about 2 minutes.
To assemble the dish: Put a puddle of the blackberry sauce on one side of the plate. Place a chicken breast on top of it. Slice the sausages and put the slices from half a sausage on top of the chicken breast. Make a nice pile of the zucchini and top that with the crispy artichoke hearts. Drizzle a bit more sauce over the chicken and sausages if you like. Repeat with the remaining ingredients.