Preheat the oven to 400 degrees F.
Wrap the beet tightly in foil. Roast the beet directly on the oven rack for 40 minutes, or until just tender when pierced with a knife. Unwrap, cool to room temperature, and then refrigerate until completely chilled. (The beet can be made 1 day ahead. Wrap in plastic and refrigerate.)
For the chile oil: In a small saucepan, combine the oil and minced chile, and set over medium-high heat. As soon as the chile begins to sizzle and turns orange in color, turn off the heat. Transfer the chile oil to a small bowl and cool completely. (The chile oil can be made 1 day ahead. Keep covered at room temperature.)
Carefully peel the beet. Using a mandolin, set at 1/8-inch thickness, or a very sharp knife, cut the beet into paper-thin rounds. Arrange the beet slices overlapping on a large platter. Sprinkle the vinegar all over. Drizzle some of the chile oil (with the minced chile) all over the beets. Season the beet slices generously with salt and pepper. Sprinkle all over with the Gorgonzola cheese. Garnish with the arugula and serve.