A butcher cuts the eye section of a beef rib roast from a single muscle after he removes the rib bones. This results in one of the choicest meats for roasting. The Delmonico Restaurant in New York popularized the rib-eye roast in the late 19th century and it's been a favorite of chefs and diners ever since. Slow-cooking a ribeye simply means that you're cutting back on your oven temperature and increasing roasting time so your beef retains more of its moisture content. You can achieve this in a crock pot, or you can do it in the oven.
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Preheat your oven to 325 degrees F, approximately 25 degrees less than you would use for regular roasting.
Prepare an herb rub for your roast while your oven is heating. Epicurious recommends combining thyme, sage, rosemary, salt and pepper. If you add minced fresh garlic, the moisture will help hold your rub together and it will stick to your roast more easily.
Rub your roast with the herb mixture and place it on a rack in a roasting pan large enough to hold it. Lay it flat side up and insert a meat thermometer into the thickest, middle part.
Roast your rib-eye for approximately 25 minutes per pound. This comes out to about two hours for a 5-lb. roast. The meat thermometer should read 140 degrees F for rare, 160 degrees F for medium and 170 degrees F for well-done when the roast is finished cooking.
Remove your roast from the oven and place it on a safe surface to rest for 10 to 15 minutes. Beef continues cooking during this process.
Crock Pot Method
Prepare stock for your roast to cook in. Cooks.com recommends using chopped onion, a packet of dry beef gravy mix, a cup of water, ½ cup catsup, 2 tsp. Dijon mustard and ½ tsp. garlic powder. You can also add wine, if you like, either in place of the water or in addition to it.
Place your rib-eye roast in the crock pot and pour the stock mixture over it. Cover tightly and set the crock pot to low.
Cook your roast up to eight hours at the low setting.
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Crock pots are ideal for tougher cuts of meat, and a rib-eye roast is very tender. If you cook a rib-eye too long by this method, with the liquid, you risk it literally falling apart when you try to take it out of the pot.