In a large sauté pan, over medium heat, heat 2 tbsp. oil until melted and beginning to froth. Add mushrooms and cook, stirring frequently for 2-3 minutes. Add garlic, sprinkle with sea salt and pepper, and continue to cook until most of the liquid has evaporated, 4-5 minutes more. Add 2 tbsp. chopped sage and cook 1 minute more. Remove from heat.
Remove skin from turkey breast and set aside (do not discard). Lay turkey breast on a work surface and cover with plasticwrap. Pound with a kitchen mallet to an even ½ to 1-inch thickness. Spread mushroom mixture evenly over turkey breast, leaving a 1-inch border and roll up from the short end tightly into a log. Wrap reserved skin over log and tie at several intervals with kitchen twine.
Place on a broiling pan and rub all over with remaining oil. Sprinkle with remaining sage and season with salt and pepper. Place pop-up timer, if using, into thickest part of the turkey roll (pierce the skin with a sharp knife if need be). Roast until timer pops up or until internal temperature reaches 180 F, approximately 50 to 60 minutes.