Heat oven to 400 degrees. Use a slicing knife and your fingers to remove skin from breast, reserving skin. Turn the breast over (so the side that had the skin is facing down) and lay it flat on the cutting board. Holding the blade of the knife parallel to the board, about halfway down, slice into the thickest portion of the breast. Cut along the length of the breast, but not all the way through. Unfold so the turkey opens like a book. Remove the tough piece of cartilage. Cover with a piece of plastic wrap and pound with a meat mallet until the turkey is a uniform thickness (about 1/2 inch). Season with salt and pepper.
Spread stuffing evenly (about 3/4 inch thick) over turkey, leaving a 1-inch border. Starting with one short end, roll into a log, completely enclosing the stuffing, and wrap the reserved skin around the breast, over the seam. Roll in a piece of cheesecloth and secure both ends with kitchen twine. Tie twine around the roast in four evenly spaced intervals, and rub butter evenly all over cheesecloth.
Roast on a rimmed baking sheet until an instant-read thermometer inserted in the middle registers 155 degrees, 70 to 80 minutes. Let rest for 10 minutes (the internal temperature should rise to 165 degrees).
Remove cheesecloth and twine, then place turkey on a cutting board and slice crosswise about 3/4-inch thick. Serve garnished with oregano.