Combine the miso, sake, sugar and scallions in a small saucepan over medium heat. Bring to a simmer and cook, stirring, until thick, about 5 minutes. Let cool, then transfer to a large bowl and whisk in the vegetable oil.
Pat the quail dry. Add to the bowl with the miso mixture and toss; cover and refrigerate 1 hour.
Preheat the broiler. Line a shallow baking pan with foil. Arrange the quail in the pan in a single layer. Broil, rotating the pan as needed, until the quail are browned in spots and cooked through, about 8 minutes per side.