In a large bowl or wide deep dish combine buttermilk, dry rub, salt, and pepper, and stir to combine. Place chicken into buttermilk bath, cover with plastic wrap, and place in fridge to marinate for 2-3 hours.
Remove chicken from the marinade. Place on a drying rack (we used a drying rack on top of a foil lined cookie sheet pan). Place back in fridge, uncovered, for one hour.
For the Apricot Glaze:
In a small saucepan over medium heat sauté shallots and jalapeños in olive oil for 8 minutes or until translucent. Add garlic and let cook 1-2 minutes. Add jam and honey, stir, and bring to a slight simmer. Reduce heat and simmer for 5 additional minutes.
Remove from heat, stir in the lime, and let cool at least 15 minutes to set and thicken.
To Grill the Chicken
Prepare grill for direct/indirect cooking. (If using a gas grill preheat grill to medium heat.)
Place chicken pieces on grill over direct heat, but be careful of flare-ups. After 4-5 minutes (or when a nice crust starts to develop) turn chicken over and allow it to cook an additional 4-5 minutes per side. When the other side has developed a nice crust, move the pieces to indirect heat to avoid flare-ups.
As the chicken nears appropriate finish temperature (165 degrees for breast, 180 for thighs) begin to brush chicken pieces liberally with the glaze.Apply all of the glaze between the chicken pieces, top side only!
Allow the chicken to finish cooking to temperature (165 degrees), and then remove from the grill. Let the chicken pieces rest 10 minutes to allow juices to redistribute. Serve