They do. I've done Eggs a few ways now.
Method one: Just put them on with another hot smoke for 2 hours. Pull, ice water, shell them, you got hard boiled eggs, but the shell is -super- easy to take off.
Method Two: 2 hours in, pull, ice water, peel, back on the smoker <And make sure to oil what ever grate you put them on.> For 30 to 60 mins in smoke. Now you got hard boiled eggs that taste smokey. If you use Hickory, they actually get a bacon like flavour profile.
Method three: Hour to two hours on, bring in, ice water. Lightly crack the shell at random, back in the smoker for an hour or so in smoke. Then bring them in, ice water and fully peel. You get dark lines through out the white where the cracked shell lines were, looks kinda cool.
Also smoking the eggs makes them -alot- easier to peel, if you run them in a smoker at 240-250f for 2-3 hours, the shell comes off darn easy after they get the ice water soak. I like to use smoked eggs in both egg salad and devilled eggs. That faint bacon like smoke flavour they can develop just takes every thing up a notch. They also take up little space in the smoker so normally I just do a dozen with any smoke. Didn't this time because I just did less then a week ago.
This was my first 'cold' smoke here-
https://www.smokingmeatforums.com/threads/unstuffed-smoked-beef-pepperoni-sticks-with-qview.95395/
I used Bear's recipe and now I've also done summer sausage and turkey pepperoni. I am actually always on the look for pork loin now, I want to find a reasonable price for one with a nice fat layer on it to make Rashers like ya'all got in the UK. American streaky bacon is good for wrapping stuff that goes on a grill or smoker, but I absolutely love rashers. Only one place has them here, and they ain't cheap by a long shot, so I figured making my own is a better option!