Iranian Rice Cookies

Preheat oven to 350°F. Line a fine-mesh sieve with 4 layers of cheesecloth; set over a small liquid measuring cup. Melt butter in a small saucepan over medium-high heat until beginning to boil. Reduce heat to medium and simmer until foamy, stirring occasionally, until butter turns golden brown with a nutty aroma, and milk solids separate into brown specks that sink to bottom, 5 to 7 minutes. Remove saucepan from heat and strain butter through prepared sieve, leaving solids behind. Let cool slightly.
In another small saucepan over medium heat, stir together sugar and 2 tablespoons water until sugar has dissolved; transfer to a large bowl. Let cool slightly. Add rose water and stir to combine.
Add egg yolk to sugar mixture; whisk until light and slightly thickened. Continue whisking and drizzle in strained brown butter until thick and incorporated. In a medium bowl, whisk together both flours, the cardamom, and salt. Add to brown butter mixture and stir until incorporated.
Roll or scoop dough into 11/2‑inch balls and place about 1 inch apart on parchment-lined baking sheets. Imprint each ball with the grooved side of a meat mallet, pressing to 1/4 inch thick. Dust mallet with rice flour as needed to prevent sticking. Bake, rotating sheets halfway through, until edges of cookies just turn light golden, 16 to 18 minutes. Transfer cookies to wire racks to cool completely. (Cookies can be stored in an airtight container at room temperature up to 3 days.)