In a blender, puree one-quarter of the strawberries, the shallot, 2 1/2 tablespoons balsamic vinegar, 2 tablespoons oil and the honey; season with salt and pepper.
Season the chicken breasts with salt and pepper. In a large skillet, heat the remaining 1 1/2 tablespoons oil over medium heat. Add the chicken and cook, turning once, until cooked through, about 8 minutes; transfer to a plate and tent with foil.
Meanwhile, add the remaining strawberries, 1 tablespoon balsamic vinegar and 1/4 cup water to the skillet and simmer, stirring, until the berries are warmed through, about 1 minute.
In a large bowl, toss the spinach with the strawberry vinaigrette and divide among 4 plates. Slice the chicken breasts 1/4 inch thick and place on top with the warm strawberries, goat cheese and chives; season with salt and pepper.