Preheat the oven to 180 C / Gas 4. Place the chicken breasts between two resealable freezer bags on a solid, level surface. Firmly pound the breasts with the smooth side of a meat mallet or heavy pan to a thickness of 5mm; set aside.
Melt the butter in a frying pan over medium heat. Stir in the almonds, and cook until dark, golden-brown, about 10 minutes. Set aside until cool enough to handle.
Lay the chicken breasts out on a board and evenly divide the toasted almonds among them. Sprinkle with the Parmesan cheese, then arrange the spinach leaves over top. Roll each chicken breast up and secure with cocktail sticks. Pour half of the vinaigrette into a casserole dish and place the chicken rolls into the dish seam-side-down. Pour the remaining vinaigrette over the top.
Cook the chicken breasts until no longer pink in the centre and the juices run clear, about 45 minutes. Let the chicken rest 5 to 10 minutes before serving.