Position an oven rack in the top of the oven, and preheat to 350 degrees F. Put the sweet potatoes on a rimmed baking sheet, and bake until fork-tender, 40 to 50 minutes. Let cool completely. (The potatoes can be baked, cooled and refrigerated a day ahead.) Split each in half lengthwise, and scoop out most of the flesh, leaving about a 1/4-inch border all around. (Save the scooped-out flesh for making mashed sweet potatoes later.)
Preheat the oven to 400 degrees F. Put a rack on a rimmed baking sheet. Arrange the potato skins skin-side up on the rack, and brush with 2 tablespoons of the oil. Bake until the skins are slightly browned, 20 to 30 minutes. Let cool completely, then cut each in half crosswise.
Arrange the pieces skin-side down on the rack, and sprinkle each with mozzarella, Parmesan and scallions. Bake until the mozzarella melts, 8 to 10 minutes. Remove the potato skins from the oven, and drizzle with the remaining tablespoon of oil. Top each with some salsa verde and avocado. Sprinkle with crushed chips.
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