Add the chicken to a 6-quart slow cooker. Sprinkle the salt evenly over the chicken to coat.
In a medium bowl, whisk together the peanut butter, coconut milk, curry paste, sugar, lime juice, and fish sauce until very smooth.
Pour ⅓ of the satay sauce over the chicken, tossing gently to coat. Set the remaining sauce aside.
Cover and cook on high for 2-3 hours, or low for 4-5 hours, or until the chicken is cooked through and very tender but not falling apart.
When the chicken is cooked, uncover and carefully drain most of the cooking liquid. Pour over half of the remaining satay sauce, tossing to coat.
Optional, but recommended: Place the cooked and sauce-coated chicken on a broiler safe pan or baking sheet. Place under a broiler for 3-5 minutes or until browned, then flip and broil 2-3 more minutes to brown the other side. Keep a close eye on the chicken, as the sugar in the sauce can brown very quickly.
Serve the sauce coated chicken with the remaining satay sauce on the side for dipping. Garnish with cilantro, fresh lime slices, and peanuts if desired.
Chicken Breasts: If using chicken breasts, you may need to add an additional hour, so 3-4 hours or high, or 5-6 on low.
Curry Paste: Use 1 tablespoon red curry paste for a milder satay sauce, and 2 tablespoons (or more!) for a spicy satay sauce.
No Fish Sauce: Sub tamari or soy sauce.
Coconut Milk: Give the can a good shake before using, as the fat often separates from the coconut water. If shaking doesn't work, use a fork to whisk it together before making the sauce.