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Preparation Place a rack in middle of oven; preheat to 350°F. Toss beets, garlic, chile powder, oregano, paprika, cumin, pepper, 1 Tbsp. Diamond Crystal or 1½ tsp. Morton kosher salt, and ⅓ cup oil in a medium bowl until beets are coated. Scrape onto a foil-lined rimmed baking sheet; bake, tossing every 15 minutes, until beets are tender and spices are very fragrant and smell slightly toasted, 35–45 minutes.
Meanwhile, heat 2 cups oil in a large saucepan over high until oil bubbles immediately when the edge of a tortilla touches the surface. Carefully fry onion, adding a little bit at time to avoid too much spattering and reducing heat as needed, until tender and beginning to brown, about 5 minutes. Transfer to a paper-towel-lined baking sheet. Repeat process with chiles, about 5 minutes. Transfer to same baking sheet. Fry sage leaves until crisp, about 30 seconds. Transfer to baking sheet. Working one at a time, fry tortillas, turning once, until crisp, puffed in places, and deep golden brown, about 1 minute per side. Transfer tostadas to baking sheet.
Heat remaining 3 Tbsp. oil in a large nonstick skillet over medium-high. Crack eggs into skillet, leaving space around each one, and cook until whites are set and edges are crisp, about 4 minutes. Season with salt.
Spoon beet mixture onto tostadas, dividing evenly, and top each with an egg, then fried onion, chiles, sage, and Salsa de Árbol.