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When ready to cook, start Traeger on Smoke with lid open until fire is established (4-5 minutes) increase temperature to 425 degrees F and preheat with lid closed for 10 to 15 minutes.
Chop the broccoli and place on parchment lined sheet pan. Drizzle with olive oil and sprinkle with a little salt.
Place pan in Traeger and cook for 5 minutes. Stir and continue cooking for 5 more minutes. Remove from grill and set aside.
Place a large cast iron pot on Traeger and allow to preheat with lid closed for 5 minutes. Add 1 tbsp butter, the diced onion, and sauté stirring frequently until onion is translucent and barely browned, about 4 minutes.
Add the garlic and cook another 30 seconds, stirring constantly so it doesn't burn. Remove from heat, take mixture out of pot and set aside. Wipe pot out with paper towels.
Turn temperature down to 375 degrees F. Place cast iron pot back on Traeger and allow to preheat again.
Add 4 tbsp butter, flour and cook with lid closed for 20 minutes whisking every 5 minutes or until flour is thickened.
Add the chicken stock, whisking constantly. Slowly add the half-and-half, whisking constantly.
Cook for 45 minutes or until mixture has reduced and thickened some. Whisk intermittently to re-incorporate the 'skin' that inevitably forms, this is normal.
Add the broccoli, carrots, onion and garlic. Add the salt, pepper, paprika, dry mustard powder, and cayenne.
Allow soup to simmer for 20 to 25 minutes more, or until it has reduced and thickened some again. While soup simmers, grate the cheese.
After simmering about 20 to 25 minutes, add most of the cheese, reserving a small amount for garnishing bowls. Stir in the cheese until melted and incorporated fully, less than 1 minute.
Serve immediately. Enjoy!