Combine the ricotta, 1/4 cup Parmigiano-Reggiano, sage and milk and season with salt.
Butterfly each chicken breast, cutting horizontally into the thickest part of the breast almost all the way through to the other side (keeping the breast attached on one side). Open each breast like a book and pound to 1/8 to 1/4-inch thick using the flat side of a meat mallet. Sprinkle the chicken with salt and pepper. Place a spoonful of the ricotta mixture in the center of each chicken breasts, then fold the chicken over to enclose the filling. Wrap each chicken breast with 2 slices prosciutto, overlapping them slightly. Wrap each chicken breast in plastic wrap and refrigerate overnight.
Preheat the oven to 400 degrees F.
Unwrap the chicken breasts and place on a rimmed baking sheet or shallow baking dish. Drizzle with EVOO, then bake until the chicken is cooked through and the prosciutto is crisp, 20 to 25 minutes.
While the chicken bakes, heat the butter in a small skillet over medium heat. Add the garlic and cook, stirring, for 2 minutes. Add the panko and thyme and cook, stirring, until the panko is golden. Remove from the heat and stir in the remaining 1/4 cup Parmigiano-Reggiano.
Divide the chicken among plates. Arrange the arugula alongside the chicken and top the arugula with the tomatoes and garlic breadcrumbs.
Cook's Note: The stuffed chicken breasts can be refrigerated for up to 2 days before cooking.