Place all of the pepper aoli ingredi- ents in a large bowl and whisk thor- oughly. Place aside in fridge.
Slice off the stem of the brussels sprouts, cut in half (then as thin as possible) and place in a bowl. Slice the green apple into thin slices and place it in the same bowl as the sprouts. Add salt and pepper. Add juice of half of a lime and toss.
Place slaw on the bottom bun. Spread aioli on the top bun.
Heat turkey broth on the stove. Place turkey in the broth for 30 seconds and remove.
Place the turkey on a metal pan. Place cheese on top of turkey and place under the broiler until the cheese melts.
Remove and place it on top of the slaw. Place the top bun and serve.