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To roast the sweet potatoes: Preheat the oven to 400 degrees Fahrenheit with two racks placed near the middle of the oven, leaving a few inches in between the racks. Line two large rimmed baking sheets with parchment paper to prevent sticking.
Place the sweet potatoes on one of the prepared baking sheets, then drizzle the olive oil on top and sprinkle with the chili powder and several dash of salt. Toss until the sweet potatoes are lightly and evenly coated with oil and spices.
Bake until the sweet potatoes are tender and caramelized on the edges, about 30 to 35 minutes. Set aside, but leave the oven on for the crispy tortillas.
Meanwhile, to cook the refried beans: In a medium saucepan over medium heat, warm the olive oil until shimmering. Add the garlic and cumin and cook, while stirring constantly, until fragrant, about 30 seconds.
Add the drained beans, water and salt. Let the mixture come to a simmer and cook for 10 minutes, stirring often, and reducing heat as necessary to maintain a gentle simmer. Remove from the heat and mash the beans with a potato masher, pastry cutter or fork until the mixture is thick and spreadable. Cover and set aside.
To prepare the crispy tortillas: On the remaining prepared baking sheet, brush both sides of each tortilla lightly with oil. Clear off the other baking sheet by transferring the roasted sweet potatoes to a serving bowl, but reuse the parchment paper. Arrange 4 tortillas in a single layer across each pan. Bake for 10 to 12 minutes, turning halfway, until each tortilla is golden and lightly crisp. Set aside.
To prepare the salad: In a medium serving bowl, combine the chopped lettuce, feta, about 1/2 cup of the red onion (reserve the rest for garnish), olive oil and lime juice. Toss to combine, and add a pinch of salt if the flavors don’t quite sing yet. Divide the salad between 4 low, wide salad bowls or dinner plates.
To assemble the tostadas, spread black beans over each tortilla, and place each one on a bed of salad. Top each tortilla with a few slices of avocado, followed by the sweet potatoes, and a sprinkle of red onion and cilantro. Serve with your favorite hot sauce or salsa on the side.
Tostadas are best consumed promptly. If you have leftovers, store individual components separately, if possible. Leftover black beans make a good side, and they’re also a great quesadilla filling.