Preheat oven to 425°F (220°C). Line a 13x9-inch rimmed baking sheet with parchment paper.

In a large bowl, whisk together flour, sugar, baking powder, salt, and baking soda. Add cold butter, tossing to coat. Using a pastry blender or 2 forks, cut in cold butter until mixture is crumbly and butter is pea-size. Stir in ¾ cup (85 grams) cheese, green onion, and jalapeño. Add cold buttermilk, and stir with a fork until a shaggy dough forms.

Turn out dough onto a lightly floured surface, and pat into a 9-inch square (about 1 inch thick). (Dough will be crumbly.) Using a bench scraper, cut dough into fourths. Stack fourths, and pat or roll into a 9-inch square. Repeat procedure twice. Gently pat dough into a 12x7½-inch rectangle. Using a bench scraper or knife, cut rectangle into12 (3x2½-inch) rectangles. Place ¼ inch apart on prepared pan. Freeze for at least 15 minutes.

Brush tops of dough with melted butter.

Bake for 15 minutes. Sprinkle with remaining ¼ cup (28 grams) cheese, and bake until an instant-read thermometer inserted in center registers 200°F (93°C), 10 to 15 minutes more. Sprinkle with green onion. Let cool on pan for 5 minutes. Serve warm with softened butter.