Remove the chicken tenders, if still attached, and trim any excess fat from the chicken breasts. With a sharp knife, cut the breasts lengthwise into 1/2-inch slices (you should have about six slices per breast). Cut each slice in half crosswise to make about 32 pieces total. If you have tenders, cut those in half, too.
Combine 1 Tbs. of the soy sauce and 1 Tbs. of the lemon juice with the oil, garlic, curry powder, 1/2 tsp. salt, and a few grinds of pepper in a medium bowl. Add the chicken and toss well to coat. Let the chicken marinate at room temperature for least 15 minutes and up to 1 hour.
Meanwhile, combine the remaining 1 Tbs. soy sauce and 1 Tbs. lemon juice with the peanut butter, coconut milk, brown sugar, cayenne, and 1/8 tsp. salt in a small saucepan.
Position an oven rack as close to the broiler as possible and heat the broiler to high. Thread one chicken piece onto the end of each skewer. Set the skewered chicken on a broiler pan and broil, turning the skewers once halfway through, until the chicken is lightly browned and cooked through, about 7 minutes total. While the chicken cooks, warm the sauce gently over medium-low or low heat. If the sauce seems very thick, thin it with about 1 Tbs. water. Let the chicken cool slightly and then serve the satés with the peanut sauce for dipping.
You can marinate the chicken, refrigerated, for up to 24 hours.