Add the butter, icing sugar, flour, salt, Earl Grey tea leaves, and vanilla extract into a food processor.
Pulse until the dough comes together.
Divide the dough in half and roll each half into a log about 1 ½ to 2 inches in diameter.
Wrap each log in plastic wrap and refrigerate for at least 1 hour.
Preheat the oven to 350 degrees Fahrenheit.
Use a sharp knife to slice the dough into ¼ to ½ inch thick rounds.
Place the cookies on a parchment-lined baking sheet, at least 1 inch apart.
Bake for about 10 minutes, until the bottoms are barely golden.
Transfer cookies to a cooling rack and let cool completely before serving.