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Combine the flour and 2 teaspoons of the Traeger Pork and Poultry Rub in a small shallow dish.
Wet the eggs with water and roll in the seasoned flour to lightly coat. Wet your hands and divide the sausage into 4 equal-size balls. Flatten each ball into an oval.
Cover each egg in sausage, smoothing the seams to encase the egg evenly. Rewet your hands as necessary.
Dip each sausage-covered egg in the beaten egg, letting any excess drain off, and then dredge in the bread crumbs.
Refrigerate the eggs, uncovered, for 1 to 2 hours. When ready to cook, start the Traeger grill on Smoke with the lid open until the fire is established (4 to 5 minutes).
Arrange the eggs on the grill grate and smoke for 1 hour. Remove from the grill and set aside. Increase the grill temperature to 375. When the grill reaches the set temperature, return the eggs to the grill grate and bake for 30 minutes. Let cool for a few minutes, then slice each egg into quarters with a sharp knife.
While the eggs bake, combine the mayonnaise, mustard, and remaining 2 teaspoons of Traeger Pork and Poultry Rub in a small mixing bowl. Line a serving plate with leaf lettuce. Arrange the quartered eggs on the serving plate accompanied by the dipping sauce, and if desired, pickles.