Though I loved pulled beef, it's not something I do with chuck eyes (nor chuck top blades). Those I prefer to roast to medium rare and slice thinly for roast beef. If that's an interest, salt and pepper the roast then brown in hot oil (I use evoo) in a pan. Remove the roast, sprinkle on the rub of choice, if desired, then move to the kettle and slow cook at ~250 to an internal of ~115, indirect. Meanwhile, when the roast nears 105 or so, light a chimney of fuel.
When the roast hit the 115 area, quickly remove the kettle lid and dump the lit fuel, splitting it half and half on the sides, through the grate flaps (if you have a solid grate, remove it and the meat and dump the fuel, returning the grate with the meat). Continue cooking at the highest temp you can muster, still indirect of course, till the roast reaches 130; remove, tent with foil and rest 15-20 min; slice thinly to serve.
After you brown the meat and it is in the kettle, add a little unsalted butter to the pan you brwoned the meat in and saute a finely chopped shallot or two, or 1/2-3/4 c finely chopped onion, over moderate heat, till soft. Add some fresh or dried thyme and a bay leaf while the onion cooks. When it is soft, add a minced or pressed clove of garlic and 1/2-3/4 c wine (I use sauv blanc here, usually) and increase the heat to med-high. Deglaze the pan, scraping up the browned bits, and reduce the wine to 1 T. (If you have any brady or cognac flame a T or 2 at this point.)
Add 1 T Dijon and whisk very well. Add 2 t Worcestershire and 3/4 c homemade beef stock (or 1/2 c canned low-salt chicken stock plus 1/4 c canned beef stock) and bring to a simmer, whisking well. Remove the bay leaf and reserve off heat.
When the roast is done and resting, heat the reserved sauce over low heat in a flat saute pan. Just before slicing, drain any accumulated juices from the rested meat into the pan, stir, then stab a T of butter on a fork and stir the sauce with the fork, allowing the butter to slowly melt and emulsify with the sauce. Repeat with another piece of butter, turn off heat under the sauce, adjust sauce seasoning, slice the beef thinly; serve, passsing the sauce on the side.