Preheat the oven to 350 degrees . In a large skillet, melt the butter over medium heat. Add the onions and cook, stirring, until softened, about 7 minutes. Remove from the heat, then stir in the sage and 1/4 teaspoon salt.
On a floured surface, divide the biscuit dough into three balls. Roll each ball into a 20-inch-long strip. Transfer the strips to a baking sheet; braid, tucking the ends under.
In a small bowl, beat the egg with 1 teaspoon water. Brush the braided dough with the egg wash and sprinkle half of the onion mixture on top. Bake for 15 minutes, then sprinkle with the remaining onion mixture. Bake until golden, about 20 minutes.