In a medium saucepan, melt the unsalted butter over medium heat, stirring occasionally. After about 5 to 7 minutes, the butter will turn golden brown and develop a nutty aroma. Add a sprig of fresh rosemary while the butter cooks. Once done, remove the rosemary and transfer the butter to a bowl to cool in the refrigerator for 30 to 45 minutes.

In a large mixing bowl, combine the chilled brown butter with 1 and 1/3 cups of granulated sugar. Using a handheld mixer, beat them together on medium speed for about 2 minutes until light and fluffy. Add in one large egg, an extra egg yolk, and 1 teaspoon of vanilla extract, mixing on low for another minute until well incorporated.

In a separate bowl, sift together 2 and 1/2 cups of all-purpose flour, 1 teaspoon of baking soda, 1/2 teaspoon of baking powder, 1/4 teaspoon of salt, and 2 teaspoons of chopped rosemary. Gradually mix this dry blend into the wet mixture until just combined, being careful not to overmix.

Using a cookie scoop, portion out the cookie dough onto parchment-lined baking sheets, ensuring they are spaced apart. Once you've scooped all the dough, place the sheets in the refrigerator to chill for at least 30 minutes.

While the dough chills, prepare the lemon sugar by combining the zest of 2 teaspoons of lemon with 1/2 cup of granulated sugar in a small bowl. Massage the lemon zest into the sugar using your fingers.

Preheat your oven to 350°F (175°C). Take the chilled cookies and roll each one in the lemon sugar mixture, ensuring an even coating. Arrange the cookies on the baking sheets about 2 inches apart, then bake for about 12 minutes.