Grind the tea: Pulse Earl Grey leaves in a spice grinder until finely ground. Do not over grind into dust.
Brown the butter: In a light colored saucepan over medium heat, melt 1 cup butter, swirling occasionally. Cook until amber, brown specks form, and it smells nutty. Immediately remove from heat and pour into a large bowl. Cool for 10-15 minutes.
Combine dry ingredients: In a medium bowl, whisk together flour, baking powder, baking soda, salt, and ground Earl Grey tea.
Cream butter and sugar: To the cooled brown butter, add 1 cup granulated sugar. Beat on medium speed for 3-4 minutes until light and fluffy.
Add egg and vanilla: Add room temperature egg and vanilla bean paste. Beat until fully incorporated and smooth.
Integrate dry ingredients: With mixer on low, gradually add flour mixture. Mix just until combined. Do not overmix.
Make lavender sugar: In a small bowl, combine ½ cup granulated sugar with ½ tablespoon culinary lavender. Stir to combine.
Shape and coat cookies: Scoop tablespoon-sized dough balls. Roll each in lavender sugar to coat. Place on parchment-lined baking sheets, 2 inches apart.
Bake: Preheat oven to 350°F (175°C). Bake for 10-12 minutes until edges are set and just beginning to golden. Cool completely on baking sheet before frosting.
Make frosting: In a medium bowl, beat softened cream cheese and butter until smooth. Gradually add powdered sugar, beating until incorporated. Add vanilla bean paste and optional purple food coloring. Beat until fluffy.
Frost cookies: Once cookies are completely cool, spread or pipe frosting onto each cookie. Let set slightly before serving.