Last time I bought an entire eye of round roast from the supermarket, I divided it into 1 roast (see my post on cooking an eye of round roast), and 6 steaks. I thought that I would need all six steak to perfect my recipe, but the first time was the charm. The problem I faced when cooking these steaks is that they are not the most tender steak on the cow. They come from the round section. It also a pretty lean steak. So overcooking the steak was my number one concern. I decided that I need to cook this thing as fast as possible with as much firepower as possible (at least possible in my kitchen). Here is what I did.
First I took two eye of round steak and marinade them in some worchestershire sauce for about 2 hours. Then I remove the steaks from the marinade, patted them dry with a paper towel, added some a couple heavy pinches of kosher salt and let them reach room temperature. As that was going on, I set my oven to broil and placed inside my cast iron grill pan (see picture at the bottom). I felt this was the best way to cook the steak as quickly as I could.
Once the steak was at room temperature and the oven was ready, I placed the steaks on the cast iron grill pan, shut the door, waited for 2 ½ minutes, flipped the steaks, waited another 2 ½ minutes and let my meat rest. This resulted in a nice medium steak (anything beyond with this teak would be just too tough). Times will vary with the oven used and the size of the steak. Trial and error is the only real way to figure out what time works best for you (fortunately I didn't have any error!). You can always use a instant read thermometer to test if your steak is done to your liking.
One last thing: Do your mouth a favor and don't forget to cut against the grain!