Pat the eye of round roast dry with paper towels.

In a medium bowl, whisk together the thick teriyaki sauce and brown sugar until most of the sugar dissolves and the mixture looks smooth and glossy.

Place the roast in the slow cooker with the fat side up if there is a visible fat cap, pour the sauce mixture over it, and use tongs to turn the roast once or twice so the sides are coated.

Cover the slow cooker and cook on LOW for 7 to 8 hours, until the beef is very tender when pierced with a fork.

About halfway through the cooking time, carefully spoon some of the sauce from the bottom of the slow cooker over the top of the roast, then promptly replace the lid.

When the beef is tender, transfer it to a cutting board, tent it loosely with foil, and let it rest for at least 10 to 15 minutes.

While the roast rests, turn the slow cooker to HIGH and leave the lid off; let the liquid bubble for 10 to 20 minutes, stirring occasionally, until it reduces into a shiny sauce that coats a spoon.

Slice the beef thinly against the grain, then return the slices to the slow cooker and gently turn them in the warm glaze, or arrange them on a platter and spoon the sauce generously over the top before serving hot.