A beef eye round roast comes from the back hip of the cow, along with round steak and bottom round. It gets a lot of exercise. This makes it a challenging cut to cook to a tender result, but it also means a lot of flavour for a low fat cut of meat.
So, what's the answer? First, do what you can to infuse additional flavour and tenderness into a lean cut that has a tendency to toughen. Second, prepare the meat so that it's cooking time will continue this process. This means starting on high heat and then reducing it. You can also apply an outer coating that will further flavour the meat and reduce moisture loss. Reducing the heat allows it time to cook while staying tender. As always, let it settle before carving. Last, make sure you carve thinly, across the grain.
Mix mustard, wine, minced garlic (if using), tarragon and black pepper in small bowl. Mix paprika, chilli powder, cumin, salt, pepper, wine and oil in gallon size food storage bag.
Cut narrow, deep slits in the roast all over and insert garlic slivers deeply or omit the garlic and insert fingers full of the mustard mixture into the slits or be brave and use both. You don't have to be neat, just get some of the flavours into the slits as deeply as you can. Refrigerate the rest of the mustard mixture.
Add the roast to the plastic bag, remove as much air as possible, and seal. Place in a dish (in case of leakage) and refrigerate all day or overnight. Turn occasionally if you can.
Preheat oven to 260 degrees C. Remove roast and mustard mixture from refrigerator and allow to come to room temperature. Once roast is room temperature, rub entire outside of roast with the rest of the mustard mixture.
Place the meat on a rack in an open roasting pan. Place the roast in the oven and close the door quickly. Reduce the temperature to 246 degrees C. Do not open the oven door from this time on.
Roast for seven minutes per pound, then turn off the oven and let the roast sit in the hot oven for 2 1/2 hours. Do not open the door during this time.
Remove the roast from the oven, tent with foil, and allow to sit for 20 minutes. Carve across the grain into thin slices to serve.
This is a low fat cut, so any flavour you can add will be a plus. Experiment with other flavours in the marinade. Just make you use wine or another acid as a tenderizer. Another tasty combination is lemon, garlic, rosemary, olive oil, honey, Dijon mustard and red wine. Make sure the roast is at room temperature before you put it into the preheated oven.
Don't open the oven after you've put the roast in. Don't be tempted to cook for more than nine minutes per pound. Allow to sit in the closed oven for the entire 2 1/2 hours.