In a large bowl, whisk together the flour, salt, baking powder, and baking soda. In the bowl of a stand mixer fitted with a paddle, mix the butter on medium speed until creamy.
Add the coconut oil and mix again on medium speed until smooth. Add the granulated sugar and mix again on medium until light and creamy, 3 to 5 minutes. Add the egg, yolk, and vanilla and mix again on low speed until combined. Add the flour mixture and mix on low speed until completely combined. Use a spatula to fold in any dry pieces of dough that may be lingering on the bottom of the bowl. Divide the dough in half; it can be used immediately or wrapped in plastic and refrigerated for up to 4 days (let the dough come to room temperature before rolling).
Adjust an oven rack to the middle position and preheat the oven to 350ºF [180ºC]. Line several sheet pans with parchment paper.
On a floured surface, roll out the dough somewhere between 1/8 in [4 mm] and ¼ in [6 mm] thick (the thinner the cookie, the crisper it will be, so this will depend on your preference). Use cookie cutters to cut out shapes, then slide a spatula underneath the dough and transfer the shapes to the sheet pans, leaving 1 in [2.5 cm] of space between the cookies. Chill the pans of cookies in the refrigerator for 15 minutes before baking. Dough scraps can be re-rolled and cut out again.
Bake the cookies, one pan at a time, until cooked through, 12 to 16 minutes. For a softer cookie, bake for 12 minutes; for a crisper cookie, bake longer, until light golden brown around the edges. Place the baking pans on a wire rack and let the cookies cool completely on the pans before icing. Repeat with the remaining cookies.