Preheat the oven to 350°F/175°C and line two large baking sheets with silicone baking mats or parchment paper. Set aside.
Add 1 1/4 cup of granulated sugar and 1 Tbsp lemon zest into a large bowl or the bowl of a stand mixer. Use your fingertips to massage the zest into the sugar to help release the oil in the zest. You should be able to smell the oils being incorporated into the sugar.
Add 1 cup butter into the bowl with the lemon sugar and mix on a medium-high speed for a couple of minutes, until the mixture becomes lighter in color.
Then mix in 1 large egg + 1 large egg yolk, 2 Tbsp of lavender milk, 1 tsp vanilla extract, and a small drop of yellow gel food coloring (optional) on a medium speed until fully incorporated. Scrape the sides and bottom of the bowl as needed with a rubber spatula.
Add 2 1/2 cups flour (use a scale or fluff the flour in the bag, then spoon it into the measuring cup and level it), 2 tsp baking powder, and 1 tsp fine salt and mix on low speed until it is JUST incorporated. Scrape the sides of the bowl with a rubber spatula.
Add 1/2 cup of granulated sugar and 1 Tbsp lemon zest into a shallow bowl and again use your fingertips to massage the zest into the sugar.
Use a 1 1/2 Tbsp cookie scoop to scoop the dough into balls (you should end up with about 30 cookies), then roll them in sugar until they're evenly coated. Place the cookies about 3 inches apart on the prepared baking sheets.
Bake for 10-11 minutes, or until the edges of the cookies are set but the centers are still a bit soft. Cool for 15 minutes on the pan, then move the cookies to a wire rack to finish cooling.
This is optional, but I love to partially dunk the fully cooled cookies in melted white chocolate and shake off any excess chocolate, then place them on parchment or wax paper to let the white chocolate firm up. Sprinkle some lemon zest on top of the white chocolate while it's still wet.
These cookies can be stored in an airtight container at room temperature for up to 5 days.