Position rack in upper third of oven; preheat to 500°F.
Cut scrubbed sweet potatoes into 1-inch-thick rounds; place in a large bowl.
Add 2 tablespoons oil, ½ teaspoon salt and ½ teaspoon pepper; toss until evenly coated.
Arrange in a single layer in a 9-by-13-inch metal baking pan (do not use a glass dish, which could shatter).
Roast, flipping once, until browned, 25 to 30 minutes.
Meanwhile, place ¼ cup butter and ¼ cup hazelnuts in a small skillet; cook over medium heat, stirring often, until the butter melts, about 3 minutes.
Add ¼ cup sage; cook, stirring constantly, until the butter is light brown and fragrant, about 2 minutes.
Remove from heat. Using a slotted spoon, transfer the hazelnuts and sage to a small bowl.
Pour the brown butter into a heat-safe bowl; freeze until solidified, about 15 minutes.
Whisk 1¼ cups broth and 1 tablespoon garlic in a measuring cup.
Carefully add the broth mixture to the pan with the sweet potatoes.
Continue roasting until the broth is reduced and the sweet potatoes are very tender, 5 to 8 minutes more.
Transfer the sweet potatoes to a serving platter.
Add the chilled brown butter to the pan; carefully swirl (or whisk) until the butter is melted and emulsified with the remaining liquid.
Drizzle the sauce over the sweet potatoes.
Sprinkle with ⅛ teaspoon flaky salt and the reserved hazelnuts and sage.
