In a dry skillet over medium heat on the stove, toast the hazelnuts for about 3 minutes, shaking the pan often. Transfer to a clean kitchen towel and let cool for about 10 minutes. Wrap and rub the cooled nuts in the towel to remove the skins (some skin will remain). In a food processor combine all of the sauce ingredients. Process to create a slightly chunky sauce. Transfer to a small bowl, cover, and refrigerate. (The sauce can be made one day ahead. Keep refrigerated.)
One at a time, place each chicken breast, smooth side down, between two sheets of plastic wrap and pound to an even ¼-inch thickness. Place the chicken, smooth side down, on a work surface. Lay three sage leaves on top of each piece of chicken and an equal amount of the cheese on top of the sage. Then wrap two slices of ham around each, staggering the ham slightly so the chicken is almost completely covered and the ends meet on the other side. Press the ham so that it adheres to the chicken. Lightly brush both sides with oil.
Prepare the grill for direct cooking over medium-high heat (400° to 500°F).
Prepare the grill for cooking over medium-high heat (450°F)..
Brush the cooking grates clean. Grill the chicken cutlets, cheese side down first, over direct medium-high heat, with the lid closed, until the chicken is cooked through and the ham begins to brown, 6 to 8 minutes, turning once after 3 minutes. To check for doneness, cut into the underside of one breast. The meat should be opaque all the way to the center. Remove from the grill and let rest for 3 to 5 minutes.
Brush the cooking grates clean. Grill the chicken cutlets, cheese side down first, over direct medium-high heat, with the lid closed, until an instant-read thermometer inserted into the thickest part of the chicken registers 165°F, and the ham begins to brown, 6 to 8 minutes, turning once after 3 minutes. Remove from the grill and let rest for 3 to 5 minutes.
Brush the cooking grates clean. Grill the chicken cutlets, cheese side down first, over medium-high heat, with the lid closed, until the thickest part of the chicken registers 165°F, and the ham begins to brown, 6 to 8 minutes, turning once after 3 minutes. Remove from the grill and let rest for 3 to 5 minutes.
In a large bowl whisk the dressing ingredients. Slowly whisk in ¼ cup oil until the dressing is emulsified. Add the greens to the large bowl and mix to coat them lightly. Divide the greens among four plates. Place one cutlet on each plate next to the greens and spoon some sauce over each cutlet. Serve warm.
