Preheat oven to 275°. In a stand mixer, beat butter and granulated sugar on medium speed with paddle attachment until well blended and fluffy. Add rice flour and beat on low speed until incorporated, scraping bowl as needed. Gradually add all-purpose flour; beat on low until blended, scraping bowl occasionally.
Line a 12- by 17-in. rimmed baking sheet with parchment paper. Press dough into pan until very even and smooth. Bake until shortbread is golden in center, 60 to 65 minutes. Reduce oven to 250°, drape shortbread with parchment paper, and continue to bake until a finger pressed lightly into center of dough leaves no imprint, 20 to 25 minutes more.
Set shortbread on a cooling rack and sprinkle with sparkling sugar. While hot, cut into 1- by 3-in. rectangles or 2-in. squares (use a wooden ruler). Let cool completely in pan. After it cools, if you find that you'd like it crunchier, pop pan into a 250° oven, covered with parchment, for about 20 minutes.
Make ahead: Airtight at room temperature, up to 1 week, or frozen, up to 3 months.
Note: Nutritional analysis is per rectangular cookie.