Sift together the all-purpose flour and kosher salt into a bowl and set aside.
In a stand mixer fitted with the paddle attachment, beat the room-temperature unsalted butter with the granulated sugar and powdered sugar until smooth and creamy.
In a small bowl, whisk the egg yolks with the whole milk and vanilla extract, then gradually add this mixture to the creamed butter, scraping the bowl as needed.
On low speed, slowly add the flour and salt mixture to the butter mixture and mix until the dough begins to come together; if it seems dry, add additional milk 1/2 teaspoon at a time until it holds.
Gather the dough into a disk, wrap tightly in plastic wrap, and chill in the refrigerator for 1 hour.
Preheat the oven to 350°F (175°C).
On a lightly floured surface, roll the chilled dough to an even thickness and cut rounds to fit mini tart pans; press each round into the molds and prick the bottoms with a fork.
Chill the formed tart shells in the refrigerator for 15 minutes, then bake on a baking sheet for 12–15 minutes, or until the edges are golden brown.
Remove from the oven and let the mini tart shells cool in the pans before removing.