Preheat the oven to 350 degrees. Line a fine sieve with four layers of cheesecloth and 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 through prepared sieve, leaving solids behind. Set aside to cool slightly.
In another small saucepan over medium heat, stir together sugar and 2 tablespoons water until sugar has dissolved, then transfer to a large bowl. Let cool slightly, then add rose water and stir to combine.
Add yolk to sugar mixture and whisk until light and slightly thickened. Continue whisking and slowly drizzle in strained brown butter until thick and fully incorporated. In a medium bowl, combine flours, cardamom, and salt. Add to brown butter mixture and stir until thoroughly incorporated.
Roll dough into 1 1/2-inch balls and place on a parchment-lined baking sheet, spaced about 1 inch apart. 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 halfway through, until edges of cookies just turn light golden, about 16 to 18 minutes. Transfer to a wire rack to cool completely.
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