In a bowl, add ginger beer, tamari, lemongrass, garlic, lemon juice and zest, brown sugar, fish sauce and half of beef broth. Place chuck roast in a one gallon zippered bag and pour marinade over top. Let sit for at least 4 hours (up to overnight).
Set smoker to 250degF. Remove roast from bag (reserve all marinade). Place directly on the grill of smoker. Smoke for two hours then flip and smoke another 2 hours.
In a 13x9 disposable foil pan, add reserved marinade and remaining beef broth. Remove roast from smoker and place in pan. Cover with foil. Increase heat on smoker to 300degF. Cook for 4 hours more.
Add all ingredients for cucumber salad to a bowl and toss to coat evenly. Cover and place in refrigerator until ready to serve.
In the last 15 minutes of cook time for the roast, add ingredients for rice to a sauce pan over medium-high heat. Cover with lid and cook for 15 minutes. Remove from heat but do not remove lid until ready to serve.
Remove roast from smoker and let sit for 5 minutes before removing roast to a bowl. Meat should be tender enough to break apart with a fork or pair of tongs. Ladle roughly a cup of the cooking liquid into the meat and stir to add some more moisture.
In a shallow bowl, add portion of rice, portion of cucumber salad and portion of beef.