Preheat the oven to 425°. Heat two rimmed baking sheets in the oven until hot, about 10 minutes.
In a large bowl, toss the potatoes with 1/4 cup oil; season with salt and pepper. Remove the baking sheets from the oven. Drizzle with oil; arrange the potatoes, cut side down, in a single layer. Roast until golden and tender, about 20 minutes.
Meanwhile, in a saucepan, melt the butter over medium. Add the garlic and cook, stirring often, until fragrant, about 30 seconds. Gradually whisk in the flour and cook, stirring constantly, until golden brown, 2 to 3 minutes. Slowly whisk in both stocks; simmer until the gravy thickens, stirring often, 12 to 15 minutes. Stir in the Worcestershire; season.
In a 2-qt. baking dish, spread out the potatoes. Drizzle with three-quarters of the gravy; top with the cheese. Bake until the cheese just melts, 2 to 3 minutes. Drizzle with the remaining gravy and sprinkle with the chives.