Using an electric mixer, beat butter in a medium bowl on low speed until smooth and fluffy, 2-3 minutes. Add sugar and vanilla. Increase speed to medium, occasionally scraping down sides of bowl, until light and fluffy, another 2-3 minutes.
In a separate bowl, combine flour, earl grey, and salt, stir with whisk to combine.
Reduce mixer speed to low and add in flour mixture, 1/2 a cup at a time, mixing only until the dough comes together. *If you are making the regular version and your dough is too dry, you may need to add water. Do so 1/2 a teaspoon at a time just until dough holds together.
Gather dough and flatten out over a dinner plate (this helps it to cool faster). Chill for around 20 minutes, or until firm to the touch. Remove from refrigerator for 5 -15 minutes, or until just soft enough to roll out.
NOTE: Dough can be made 5 days in advance before baking. Store in airtight container until ready to bake.
Arrange a rack in center of oven and preheat to 350°. Line a baking sheet with parchment paper. Roll out dough on a lightly floured surface until about 1/4-1/3 inch thick. Use a 1-2 inch cookie cutter ( or size/shape of your choice) and place cookies on baking sheet.
If making the gluten free version, bake for 12-15 minutes; if you are making the regular version, bake for 8-10 minutes. Bake until cookies are firm to the touch and just barely browning around the edges. Let cool on sheets for about 5 minutes and then remove and place on wire cooling rack. Store in airtight container for up to 5 days or freeze for 1-2 months.