Heat a grill pan over medium-high heat.
Brush both sides of the sweet potato slices with 1 tablespoon of the oil. Grill the slices on one side of the grill pan until dark grill marks appear and the slices can be pierced easily with a knife, 3 to 4 minutes a side. Grill the turkey burgers on the other side of the grill pan according to the package directions for the grill.
Meanwhile, add the kale, almonds, lemon juice, 1 teaspoon salt, a few grinds of pepper and the remaining 3/4 cup oil to a blender. Puree, scraping the sides as needed, until smooth.
Mash the avocado with a fork in a medium bowl until smooth. Fold in the mayonnaise until combined.
Spread the kale pesto on both halves of each hamburger bun, then spread the avocado mayonnaise on top of the pesto. Top the bottom half of each bun with 1 tablespoon alfalfa sprouts and a burger patty. Top the burger patties with some sweet potato slices, a tomato slice and the top bun.
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