Preheat oven to 400°F (200°C). Lightly grease a baking tray with a little of the extra butter and set the tray aside.
Cut the chilled salted butter into small cubes to help processing.
In the bowl of a food processor, combine the vanilla extract, all-purpose flour, brown sugar, egg, baking soda, salt, and the cubed chilled butter.
Pulse the mixture in short bursts until it resembles rice-sized granules and starts to come together when pressed (do not overprocess).
Transfer the mixture to a bowl and press it together with your hands to form a cohesive dough. If it seems warm or too soft, chill it in the refrigerator while the oven finishes preheating (about 10–15 minutes).
Portion the dough into balls about 1 inch (2.5 cm) in diameter. Place the balls on the greased baking tray (or on parchment), leaving about 1 inch (2.5 cm) between each.
Press each ball down gently to flatten to about 1/4 inch (≈0.5 cm) thickness.
Bake in the preheated oven until the cookies are lightly golden, about 15–20 minutes. Begin checking at 12 minutes because baking times vary by oven.
Remove the tray from the oven and transfer the cookies to a wire rack to cool completely.
Serve cooled with coffee or tea, or store in a tightly closed jar at room temperature for up to 24 hours, or refrigerated for up to two days.