In a small bowl, mash the butter, chopped cilantro, lime zest, Tabasco, and 1/4 tsp. kosher salt with a wooden spoon or rubber spatula to form a smooth butter. Scrape onto a piece of plastic and roll and twist the plastic to shape the butter into a short log. Refrigerate until firm enough to slice, 20 to 25 minutes.
Meanwhile, put the butterflied chicken in a shallow nonreactive pan or on a plate. Season with 3/4 tsp. salt and drizzle with the lime juice and olive oil. Let the chicken sit for 15 minutes.
Meanwhile, heat a gas grill to high. Lay the chicken breasts on the hot grill and cook, covered, until they have grill marks, 1-1/2 minutes. With tongs, rotate the chicken 90 degrees (to get a crosshatch of grill marks), and continue grilling until grill marks form, 1-1/2 more minutes. Flip the breasts and cook the second side in the same way but for a little less time, grilling for 1 minute in one direction and 1 minute in another, until cooked through. Immediately transfer the chicken to two warm serving plates, thinly slice the chilled butter, and arrange the butter slices all over the top of the chicken. Garnish with the cilantro sprigs and serve immediately.