For the Shallots: Preheat the oven to 300°F (150°C). On a rimmed baking sheet, toss shallots with olive oil until evenly coated. Season with salt, then arrange in a single, even layer. Roast shallots, rotating baking sheet halfway through cooking, until golden brown and crispy, 45 to 60 minutes. Set aside.
For the Beets, Carrots, and Onions: Increase oven temperature to 400°F (200°C). On a rimmed baking sheet, toss beets, carrots, and onions with the olive oil; season with salt. Spread vegetables out in an even layer and roast, stirring vegetables and rotating the baking sheet once halfway through cooking, until a paring knife can easily pierce beets and carrots, 45 to 60 minutes. Remove from heat, cover with a layer of foil, and let rest 5 minutes.
Meanwhile, for the Sweet Lime Dressing: In a small bowl, whisk the lime juice and zest, garlic, sugar, black pepper, and red pepper flakes until the sugar dissolves completely. While drizzling in a slow, steady stream, whisk in olive oil to form a creamy dressing. Season with salt.
In a large serving bowl, toss roasted beets, carrots, and onions with the dressing until evenly coated. Garnish with the sesame seeds, scallions, and crispy shallots. Serve warm or at room temperature.
