In a skillet big enough to hold the chicken in a single layer, place the chicken, apple juice (enough to just cover the chicken), salt, pepper and garlic. Bring to a simmer over medium heat. Cook, uncovered, at a bare simmer (do not boil) until the chicken is still slightly pink in the center, about 10 minutes.

Remove from the heat and let cool in the liquid (where it will continue cooking). Measure out and reserve 1/4 cup of the cooking liquid. Fish out the garlic and set aside.

Meanwhile, preheat the broiler. Broil the pepper pieces, skin-side up, four inches from the heat until the skin is completely charred, about 12 minutes. Turn the pepper pieces skin-side down and let cool on the broiler pan. When cool enough to handle, discard the skin.

In a food processor, combine the poached garlic and cashews, and process to a coarse puree. Add the skinned peppers, reserved cooking liquid, oil and vinegar, and process to a smooth puree.

To serve, cut the chicken across the grain into 1/4-inch slices and serve with the sauce.