Position a rack in the upper third of the oven; preheat to 450 degrees F. Pierce the pork all over with a fork. Combine the brown sugar, sage, garlic, 3/4 teaspoon salt and a few grinds of black pepper in a bowl; rub all over the pork.
Toss the sweet potatoes and onion with 2 tablespoons olive oil and the cayenne on a baking sheet; season with salt and black pepper. Roast, tossing once, until golden, 20 minutes; set aside.
Heat the remaining 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the pork and cook, turning, until golden, 5 minutes; transfer to the baking sheet with the vegetables (reserve the skillet). Roast until the sweet potatoes are tender and a thermometer inserted into the center of the pork registers 145 degrees F, 11 to 13 minutes.
Meanwhile, add the broth to the skillet; bring to a simmer over medium heat, scraping up any browned bits with a wooden spoon. Stir in the mustard, chives and a few grinds of black pepper. Slice the pork; drizzle with the pan sauce. Serve with the roasted vegetables.