In a large stand mixer, beat butter and sugar together until well blended. Gradually add egg yolks one at a time, then add the vanilla and salt and mix for another minute.
Add flour to the mixture slowly, mixing just until blended.
If some of the mixture at the bottom of the mixer is still crumbly at this point, you may need to knead the dough very gently just until the crumbly parts are well incorporated and softened with the rest of the dough. Divide the dough in half, and roll into cookie logs that are about 1 ¼ to 1 ½-inch round each. Wrap each cookie log in wax paper and refrigerate for at least 30 minutes.
Meanwhile, preheat your oven to 350 degrees.
After the cookie logs have chilled, remove them from the refrigerator, unwrap, and slice into thin cookies.
Place about 2 inches apart from each other on a baking sheet covered in parchment paper, or a Silpat.
Bake for about 14-18 minutes, until the edges are lightly golden. Remove and cool on cooling racks for about 5 minutes.
While the cookies are cooling, whisk together the confectioner's sugar, lemon juice, and lemon zest in a medium bowl. The final mixture should be thick but still pourable. You can add more lemon juice if it's too thick or more confectioner's sugar if it's too runny.
Dip one side of each cookie into the lemon sugar glaze and set aside, allowing the glaze to set.