The chuck eye steak comes from the upper shoulder of the cow, the region butchers refer to as the chuck primal. Known for flavorful roasts, chuck cuts generally benefit from low-and-slow cooking methods. But because the chuck eye includes a few inches of the tender longissimus dorsi muscle, the main component of a rib-eye, it can handle the high-heat cooking methods more appropriate for steak.
Tender rib-eyes come from the sixth to twelfth ribs of a cow; butchers cut the chuck eye from the fifth rib. This proximity means the chuck eye steak shares many of the characteristics of a rib-eye. Although chuck eye steaks aren't always available they tend to be a tender and robustly beefy, budget-friendly cut of meat when you can find them.
Cook chuck eye steaks quickly over high heat, as you would a rib-eye or other prime cut. Pay close attention, though—an overcooked chuck eye becomes dry and tough.
Preheat your grill or cast iron to high and brush with oil. Put the steaks on the grill or sear on stove top and cook for 4 to 5 minutes per side. For best results use a meat thermometer and take it off the grill when it reaches around 125 - 130 F. The steak will continue cooking for a final temperature of around 135 for a medium-rare finish.