In a large glass or ceramic baking dish, mash the achiote paste with the garlic and ginger. Stir in the ketchup and honey, then add the orange juice and cilantro. Set aside 1/2 cup. Add the chicken breasts to the remaining marinade, turning to coat them well. Cover with plastic wrap and refrigerate for at least 1 hour and up to 4 hours.
Meanwhile, cut the tips off of the plantain. Make a cut just through the skin, then peel. Using a mandoline, thinly slice the plantain lengthwise.
In a large nonstick skillet, heat the oil until shimmering. Working in batches, fry the plantain strips over moderately high heat until golden and crisp, 3 to 4 minutes. Using tongs, transfer the plantain strips to paper towels, letting as much oil as possible drip back into the skillet. Sprinkle the plantain strips with salt.
Light a grill or preheat the broiler. Remove the chicken from the marinade and season with salt and pepper. Grill or broil for about 10 minutes, turning, until browned and just cooked through. Transfer to a platter and cover with foil. In a small saucepan, stir the chicken stock into the reserved 1/2 cup marinade and boil over high heat until slightly reduced, about 5 minutes.