In a food processor, combine the ginger, garlic, peanut butter, rice vinegar, soy sauce, mirin (or sugar), 1-1/2 tsp. of the curry paste, and 3 Tbs. water. Process until smooth, about 30 seconds. Taste to check the heat level and add the remaining 1/4 tsp. curry paste if you like.
If the chicken breasts come with tenders, remove them and reserve them for another use. Between two sheets of plastic wrap, lightly pound the chicken breasts to an even thickness with a meat mallet, a heavy pan, or the side of a cleaver. Season the chicken all over with salt and pepper. Set a large, heavy skillet over medium-high heat and add the canola oil. When the oil is very hot, add two of the chicken breasts and cook until nicely browned, 3 to 4 min. Flip the chicken and continue to cook until it’s cooked through, another 3 to 4 min. Transfer the chicken to a platter. Repeat with the remaining chicken, adding 1 to 2 Tbs. more oil if the pan is dry.
Reduce the heat to low and add the coconut milk and 1/2 cup of the peanut sauce to the skillet. (Save the remaining sauce for another use). Stir to combine and heat through, about 2 min. Remove from the heat and stir in the cilantro. Drizzle the sauce over the chicken and serve immediately.