Begin by melting the unsalted butter in a saucepan over medium heat, swirling occasionally until golden brown and nutty, about 5-7 minutes. Add the rosemary sprig and then remove it once done. Cool the butter, then refrigerate for 30-45 minutes until it solidifies.

In a large mixing bowl, combine the chilled brown butter and white granulated sugar. Beat with an electric mixer for about 2 minutes until light and fluffy. Incorporate the large egg, egg yolk, and vanilla extract, mixing for one more minute.

In a separate bowl, sift together the all-purpose flour, baking soda, baking powder, salt, and chopped fresh rosemary. Gradually add this to the wet mixture and mix gently until just combined to maintain a tender texture.

Portion out the cookie dough onto a parchment-lined baking sheet, spacing them about 2 inches apart. Cover with plastic wrap and refrigerate for at least 30 minutes.

Mix together the granulated sugar and lemon zest in a small bowl, rubbing the zest into the sugar. Set aside while you preheat the oven.

Preheat the oven to 350°F (175°C). Roll each chilled cookie dough ball in the lemon sugar mixture, coating fully. Bake for about 12 minutes or until edges are golden and centers are puffy. Let cool completely on a wire rack.