Prepare the salad ingredients:
Heat the oven to 425°F. Season the chicken breasts with salt and pepper. Roast on a rack set in a rimmed baking sheet or roasting pan until an instant-read thermometer inserted in the thickest part of a breast registers 165°F, 40 to 45 minutes. Let cool. Remove and discard the skin and then shred the meat. Reduce the oven temperature to 375°F.
Stack the wonton wrappers on a cutting board and cut them into 1/2-inch-wide strips. Line a baking sheet with foil and spray lightly with cooking spray. Separate the strips, lay them on the baking sheet, and mist them lightly with the cooking spray. Sprinkle lightly with salt. Scrunch each strip to give it a wavy shape, if you like. Bake at 375°F until golden, 7 to 9 minutes. Reduce the oven heat to 350°F.
Spread the sliced almonds on a baking sheet and toast in the oven until golden, 6 to 8 minutes.
Bring a medium saucepan of salted water to a boil. Have a bowl of ice water ready. Boil the snow peas just until bright green but still crisp, about 20 seconds. Drain and transfer to the ice water to stop the cooking. Drain.
Put the sesame seeds in a dry skillet and shake or stir over medium heat until light golden brown, 3 to 4 minutes. Remove them from the hot pan to prevent overcooking.
Make the dressing and assemble the salad:
In a medium bowl, combine the vinegar, tamari, sweet chile sauce, garlic, ginger, salt, hot chile sauce, and pepper. Gradually whisk in the peanut and sesame oils.
In a large bowl, toss the cabbage, romaine, and snow peas. In another bowl, toss the chicken and scallions with 1/4 cup of the dressing. Add the chicken to the greens, and then add the sesame seeds and almonds. Toss with enough of the remaining dressing to coat well. Garnish each serving with the baked wonton strips.
Photo: Scott Phillips