LOL.
It's from the round, not the chuck. The chuck's in the front of the animal, and the round's on the other end.
Eye of round from a CAB is probably best roasted to medium rare (no more than 140 deg F internal -- 125 - 135 would be better), well rested, and sliced very thin. Either use the smoker at around 250, figuring abpit 40 minutes a pound; or the oven at 350 figuring around 20 minutes a pound.
Eye of round is not a good candidate for braising, or cooking extremely well done. The meat will go from dry and tough to dry and stringy without any intermediate stops.
Pan roasting is also a possibility if the roast is not too big. This is a good technique to learn. Heat a skillet (with a metal handle) very hot, add a little extra virgin olive oil and get the seasoned roast in there in a hurry. You don't want the oil to burn. Brown it on all sides, lay it flat and put some chopped onions, carrots and celery in the pan. Put the pan, meat, vegetables and all, uncovered in a 350 degree oven and cook for roughly 15 minutes a pound to medium rare. When the meat is done, remove the pan from the oven. Set the meat aside to rest, and turn the fire under the pan to hot. Add about 1/2 cup red wine and deglaze the pan by scraping all the meat juices stuck to the bottom and mixing them in with the wine as it evaporates. When the wine is mostly boiled off, add a little tomato paste and stir it around, to cook the "raw" off -- about 3 or 4 minutes. Add some Worcestershire sauce and either a cup more wine or some beef stock. Adjust the heat to a fast simmer, and reduce a little. Taste and if it's nice and beefy you're ready to adjust the salt and pepper and to thicken. If not, cook it down until it tastes good. Thicken slighltly with a flour slurry (1/2 tbs flour dissolved in 2 tbs cold water), by adding the slurry, and whisking the gravy. The raw will cook off the flour and the flour will completely absorb in about four minutes. Taste again and adjust the seasoning if necessary (it won't be). Using a fine strainer or sieve, strain the vegetables (as well as any lumps) out. Straining not only gets bad stuff out, it makes the sauce shine. It's worth the trouble.
Eye of round doesn't have a grain, so you don't have to worry about special slicing -- other than slicing very thin. You'll need a really sharp knife. If you don't have one, use the long thin slicer, third from the left on my avatar.
Luck,
Rich