Mash the unsalted butter with the parsley, dill, and garlic until uniform; season the mixture with the kosher salt and black pepper, divide into 4 equal logs (about 2 tbsp each), wrap, and freeze until very firm, about 30 minutes.
Trim the chicken breasts, boneless, skinless; remove the tenderloins and reserve as patches, then butterfly or pound each breast between plastic wrap to about 0.25 inch thickness without tearing.
Place a frozen butter log near the narrow end of each flattened breast; fold the sides over, tuck in the ends, roll up tightly around the butter, and use a reserved tenderloin to cover any thin spots or seams. Wrap each roll snugly in plastic wrap to help hold shape and refrigerate 30 minutes.
Heat the oven to 350°F. Set up a breading station with three shallow dishes: all-purpose flour in the first, the large eggs (beaten) in the second, and the plain dry breadcrumbs in the third.
Unwrap a chilled roll and dredge in all-purpose flour, shaking off excess; dip in the large eggs, then coat evenly in the plain dry breadcrumbs. Set on a rack and refrigerate 20 minutes, then repeat the coating to double-bread and seal.
Pour the vegetable oil into a large deep skillet to a depth of about 0.75 inch and heat over medium-high until 350°F (a breadcrumb should sizzle immediately on contact).
Fry 2 rolls at a time, turning carefully, until deep golden brown on all sides, 3–4 minutes per side; transfer to a rack set over a baking sheet and repeat with the remaining rolls.
Transfer the rack with fried rolls to the oven and bake until the centers reach 165°F and you see a faint bead of melted butter at an end, 12–15 minutes.
Rest 5–10 minutes to let the butter settle before serving. Slice at the table so the butter flows onto the plate.