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When ready to cook, start the Traeger grill on Smoke with the lid open until the fire is established (4 to 5 minutes). Set the temperature to 400F and preheat, lid closed, for 10 to 15 minutes.
Season the chicken breasts with Traeger Pork and Poultry Shake. Arrange on the grill grate and grill, turning once, for 15 to 20 minutes.
Remove from the grill, let cool slightly, and cut into 3/4-inch cubes. Leave the grill on.
In the meantime, make the Alfredo sauce: Melt the butter in a saucepan on the stove over medium heat. Add the garlic, and cook for 2 minutes. Add the cream and simmer the mixture for 10 minutes, or until thickened. Whisk in the 1/2 cup of grated Parmesan, the pepper, and the nutmeg. If the mixture is too thin, let it simmer a few minutes more.
Assemble the pizzas on a rimless baking sheet. Top each crust with Alfredo sauce. Divide the diced chicken and peppers between the two pizzas. Top each with 3/4 cup of mozzarella cheese and a teaspoon of dried basil. Dust the tops with additional Parmesan cheese.
Put the pizzas directly on the grill grate and bake for 10 to 12 minutes. Let cool slightly before cutting into wedges.