In a large bowl, whisk together the Buttermilk (2 cup), Dijon Mustard (2 teaspoon), Paprika (1 teaspoon), Cayenne Pepper (1/4 teaspoon), Kosher Salt (2 tablespoon), and Freshly Ground Black Pepper (1/2 teaspoon). Add the Garlic (2 clove), Shallot (1), and Jalapeño Pepper (1).
Pat the Chicken Breast (6) dry with paper towels, then add it to the bowl and toss lightly to coat with the marinade. Cover and refrigerate for at least 8 hours and up to 24.
When ready to grill, brush the grill grates lightly with olive oil. Prepare a charcoal fire with one layer of coals or for a gas grill, turn the burners to high, close the lid and heat for 10 minutes, then lower the burners to medium-high. You can also use a grill pan on the stove over medium-high heat.
Lift the chicken pieces from the marinade, place them on a large, flat plate, and sprinkle generously with salt and pepper. Grill for 7-8 minutes on each side, until lightly charred on the outside and cooked through.
Transfer the chicken to a serving platter and allow to rest for at least 10 minutes before serving. Brush any juices that have accumulated on the platter onto the breasts, sprinkle with Fresh Chives (to taste) and Coarse Sea Salt (to taste), and serve.