If using chicken breasts, halve them horizontally, making 2 thinner cutlets. In a large sealable bag or container, add the chicken, smashed garlic, sugar and pickle brine. Move the chicken around to make sure the brine is coating all the pieces. Cover and refrigerate for at least 6 hours, or up to overnight, flipping the bag (or shaking the container) every once in a while to make sure all pieces brine evenly.
Meanwhile, in a food processor (or a medium bowl) add the butter, pickles, 2 tablespoons brine, chives, dill, and garlic powder. Pulse (or mash with a fork) until fully combined, with small bits of pickle remaining. Season to taste with salt and pepper. Cover and refrigerate until ready to use. (Butter can be made up to 1 week ahead).
When ready to grill, take the butter out of the refrigerator to soften. Heat a gas or charcoal grill to high (make sure it’s fully preheated to prevent sticking). Lay the chicken onto paper towels and pat fully dry, discarding the brine. Drizzle chicken with oil and season lightly with kosher salt and black pepper.
When grill is hot, clean and oil the grates, then add the chicken and cook, flipping once. If using thighs, cook until 185°F in the thickest part, 3 to 6 minutes a side, depending on thickness. If using breasts, cook until 160ºF, 2 to 4 minutes a side.
Transfer the chicken to a platter and while hot, top each piece with a tablespoon or so of the softened pickle butter and allow to rest for 5 minutes. Top with extra chives and dill, if desired. Serve with more of the butter alongside.