Preheat the oven to 350ºF.

Fit a wonton wrapper into each cup, folding and pleating to fit. Bake until the cups are golden brown, about 10 minutes. Let cool in the pans for 5 minutes. Carefully remove from the pans and let cool completely. (The wonton cups can be stored in an airtight container at room temperature for up to 1 day.)

Season the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper. Put on a baking sheet, skin side up, and bake at 350ºF until an instant-read thermometer inserted in the thickest part of the chicken reads 165ºF, about 40 minutes. Let cool completely. Remove the chicken skin and bone and cut the meat into fine dice.

A handful at a time, squeeze the juice from the shredded ginger into a medium bowl and discard the solid ginger; you should have about 1 tablespoon of liquid. Add the lime juice, lime zest, rice vinegar, soy sauce, hot pepper, and sugar and whisk to dissolve the sugar. Whisk in the canola and sesame oils. Add the chicken, bell pepper, and 1 tablespoon of scallions and toss well. Season with salt to taste. (The chicken salad can be covered and refrigerated for up to 12 hours.)

When ready to serve, spoon the chicken salad into the cups. Divide the blackberry pieces among the cups. Sprinkle with sesame seeds and remaining scallions and serve immediately.