Make the herb butter. Combine all the butter, herbs and lemon juice ingredients in a medium-large bowl, and, using a rubber spatula, mix until well combined. Season with salt, pepper, and cayenne.
On a work surface, spread out a piece of plastic wrap, about 10 inches long. Put the herb butter across the bottom center of the plastic wrap, and form into a log-like shape, about 4 inches long. Fold the bottom edge of the plastic wrap over the butter and roll the enclosed butter forward until completely wrapped, to form a tube about 1 1/2 inches in diameter. Twist the ends together, like a party favor, and freeze until very firm.
Prepare the chicken. On a work surface, place 1 breast half between 2 layers of plastic wrap or wax paper and, using a meat-tenderizing mallet, pound to a 1/4-inch thickness. Repeat with the remaining chicken.
Unwrap the butter and cut lengthwise into four equal pieces. Lay out each breast half and season both sides assertively with salt and pepper. Place 1 piece of butter across the bottom center of each breast. Fold the bottom edge of a breast over the butter, fold in the sides, and roll the just enclosed butter forward until completely wrapped, to form a tight rolled package. Repeat with the remaining breasts.
In a large bowl, whisk together the breadcrumbs, 1 teaspoon salt, and the thyme and season generously with pepper.
Place the flour and eggs in separate flat bottomed bowls or lipped plates. Working with one chicken roll at a time, dredge it in the flour, dip in the egg, and roll in the breadcrumb mixture. Shake off the excess. Transfer to a parchment paper-lined plate. Repeat with the remaining chicken. Refrigerate for at least 2 hours and up to 12.
In a large straight-sided skillet, pour the oil to a depth of about 1/2 inch. Heat the oil over medium heat until it registers 325 degrees F on a deep-frying thermometer. Add the breaded chicken and cook, turning once, until golden brown and cooked through, about 25 minutes. Using tongs, transfer to a paper towel-lined plate.
Divide the chicken among plates and serve immediately. When cutting into the chicken, beware of spurting herb butter!
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