Place 1 stick unsalted butter in the bowl of a stand mixer (or a large bowl if mixing by hand or with an electric hand mixer). Let the butter and 1 large egg sit out at room temperature until the butter is softened.
Finely grate the zest of 1 medium lemon, then juice the lemon until you have 1 teaspoon juice. Place 1 1/2 cups all-purpose flour, 1/4 teaspoon kosher salt, and 1/4 teaspoon baking soda in a medium bowl and whisk to combine.
Add 3/4 cup granulated sugar to the butter and beat with the paddle attachment on medium-high speed until fluffy and smooth, 3 to 4 minutes. Add the egg, lemon zest, lemon juice, and 1 teaspoon vanilla extract, and beat on medium speed until well incorporated. Reduce the mixer speed to low and add the flour mixture in 3 additions, beating well after each addition to incorporate.
Cover and bowl and refrigerate at least 1 or up to 3 hours to firm up the dough.
Arrange a rack in the middle of the oven and heat the oven to 325ºF. Line 2 baking sheets with parchment paper.
Scoop out tablespoon portions of the dough and roll each portion into a ball. Using your thumb, gently press the center of each ball to flatten slightly. Place on the baking sheets, up to 12 per baking sheet with at least 1 1/2 inches between each dough ball.
Bake one sheet at a time until golden brown around the edges, 9 to 11 minutes. Let cool for 5 to 10 minutes before serving.
Store the cookies in an airtight container at room temperature up to 1 week.