Whipped Brown Butter Icing: In small nonstick skillet, melt half of the butter over medium heat; continue to cook, swirling frequently, until nutty brown and fragrant, about 3 minutes. Scrape into bowl. Let cool completely.
In bowl, beat remaining butter until light and fluffy, about 2 minutes. Beat in brown butter until combined, about 1 minute. Beat in brown sugar. Refrigerate until chilled, about 1 hour. (Make-ahead: Cover and refrigerate for up to 3 days.)
Scones: While icing is chilling, in bowl, whisk together flour, sugar, baking powder, pumpkin pie spice, baking soda and salt. Using pastry blender or 2 knives, cut in butter until mixture resembles coarse crumbs with a few larger pieces. In separate bowl, whisk together pumpkin purée, egg, buttermilk, vinegar and vanilla; using fork, stir into flour mixture just until ragged dough forms.
Turn out onto lightly floured work surface; knead about 10 times to form smooth dough. Using floured hands, gently press into 10- x 7-inch rectangle. Cut into 6 squares; halve each square diagonally to make 12 triangles.
Arrange, 1 inch apart, on parchment paper–lined rimless baking sheets. Bake, 1 sheet at a time, in 400°F oven until golden, 12 to 15 minutes. Serve with Whipped Brown Butter Icing.
Makes 12 scones.