Preheat the oven to 350°F. In a stand mixer fitted with the paddle attachment, add the butter, granulated sugar, brown sugar and salt. Turn the mixer to medium-high and beat until light and creamy (about 3-4 minutes). Stop the mixer and use a spatula to scrape the bowl.
In a small bowl, whisk the egg yolks, lemon juice, lemon zest, thyme and vanilla extract. Turn the mixer to low and add the wet ingredients, mixing until just combined. Scrape the bowl again.
Remove the bowl from the mixing stand and sift in the pastry flour, cornstarch, baking powder and baking soda. Replace the bowl to the stand, then turn the mixer to low and mix until just combined.
Scrape the dough onto a lightly floured workspace, shape into a thin disk and wrap in plastic wrap. Place the dough in the fridge for an hour.
Unwrap the chilled dough and place on a lightly floured workspace. Roll the dough to 1/4-inch thickness. Using the cookie cutter, cut as many cookies as possible and transfer them to the sheet pans lined with parchment paper, spacing them 1 inch apart. Gather the scraps of dough and reroll, cutting out more cookies and adding them to the baking sheets. (You should have about 21 cookies on each sheet). Discard the remaining dough. Place the cookies in the fridge for 10 minutes, or until firm.
Bake the cookies for 7-9 minutes, or until lightly golden brown and dry to the touch, rotating the pans halfway through the baking time.
Icing
While the cookies are baking, make the icing. Sift the powdered sugar into a medium-sized bowl. Add the lemon juice and food coloring, whisking until smooth.
Remove the cookies from the oven and let cool for 2 minutes. Brush the centers of the cookies with a light layer of icing. Once you’ve glazed all the cookies, repeat the process once more with another layer of icing. Immediately zest the lemon over the cookies. Let sit until set (about 10 minutes).