Preheat the oven to 400 degrees . Coat the onion with 1 tsp. EVOO; season with salt and pepper. Transfer to a baking sheet.
Wrap the beet in foil. Place on the baking sheet with the onion and roast until the vegetables are tender when pierced with a knife, about 45 minutes. Let the beet cool in the foil. Let the onion cool slightly, then halve lengthwise and discard the dried outer layers. Peel the beet and chop into 1/4-inch pieces.
Preheat the broiler. Spread open the onion layers slightly and return to the baking sheet. Drizzle each onion half with 1 tsp. EVOO; season and broil until the edges darken, about 3 minutes. Divide between 2 plates. Sprinkle with the blue cheese and tent with foil.
In a saucepan, bring the beet and 1/4 cup water to a simmer; season with salt and pepper. Add the butter; cook, swirling, until the sauce is thickened, 3 to 5 minutes.
Spoon the beet over the onion. Top with the walnuts, chives and a drizzle of balsamic glaze.