Preheat oven to 400 degrees.
Combine crumbs and broken-up goat cheese in a food processor. Pulse and blend to meal consistency. Transfer to a baking pan and brown mixture in oven for 5 minutes, then turn the crumb mixture with a fork and continue to brown for an additional 5 minutes.
Place a small sauté pan over medium heat; add butter and sliced onions. Caramelize the onions over medium heat for 10-12 minutes, stirring occasionally, until light brown in color. Add the balsamic vinegar and simmer until evaporated.
Place peeled raw garlic cloves in the bottom of a saucepot. Pour oil over to cover. Place over burner and bring to a low simmer for 30 minutes. Strain off oil. Let oil cool. Place soft garlic in food processor and pulse to purée.
Preheat sauté pan to medium high. Rub lamb with garlic olive oil, and then season the top of the lamb rack with salt and pepper. Sear top of the seasoned lamb rack in hot pan for 3 minutes. Place in the 400-degree oven and bake for 12-15 minutes or until medium rare. Remove and let rest for 5-10 minutes before slicing into individual lollipop chops. Smear or dip the garlic purée over the corner of each individually sliced lamb chop. Dip the coated corner into the breadcrumb mixture and plate