I was asked to smoke a venison tenderloin

Don't smoke it! Cook that puppy quickly over some pecan chunks on a grill! There's not enough fat for a traditional smoke. S&P rub with some Olive oil is good enough for my favorite meat in the world!!! pull at 130 and wrap for 1/2 hour

God I love venison tenderloin - you lucky dog
 
Season as you wish but hot and fast is the only way to go with that cut. Do not over cook or it will dry out and be nasty. As others have said pull at 130 and rest a bit. It should be a bit gray on the outter .5" and pretty red in the middle.
 
I did some back during the holidays that turned out pretty good. I injected them with melted butter then butterflied and stuffed them with cream cheese and jalapenos. Foe seasoning I used salt, pepper and a hit of cayenne. Then wrapped with bacon and smoked indirect at 275 over oak and pecan wood on my Shirley offset. I pulled at 133 internal because others were eating it as well. I would have pulled at 120 if it would have been just for me. Turned out pretty good I thought.
 
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I go the quick hot route for tenderloins. Pull them off the heat while medium rare.
For back loins (backstrap) I butterfly, stuff and wrap in bacon before smoking.
 
Hot & fast as others have noted. Hit it with your favorite seasoning(s) or rub, I like to cook mine direct until the IT hits about 120, pull it off to rest, slice and enjoy.

I've done a simple salt, pepper, garlic rub with great success as well as rubs like Oakridge BBQ's venison & wild game.

If it's the larger backstrap I have done as others have noted and butterflied, stuffed with various things (favorite is feta cheese) and grilled. Or you can cut "chops" and butterfly like these pictured below...







Feta stuffed backstrap...



 
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