Preheat oven to 350 °F. Line two baking sheets with parchment paper.
Grind tea powder with lavender flowers using a mortar and pestle or a spice grinder. If your tea is whole leaf loose tea you'll need to grind it more than the fine tea powder that comes in most tea bags (but you still need to grind up the lavender).
Measure out 1/2 teaspoon of ground tea powder and mix with 1/4 cup of sugar; you'll use this as the coating for your cookies. Set aside another 1/4 teaspoon for the glaze.
In a small saucepan or microwave safe container, gently melt butter until just barely melted. Add remaining ground tea powder (about 1 ¼ to 1 ½ teaspoons) and stir to incorporate. Let cool to lukewarm (about 90 °F or below).
In a bowl, combine sugar and brown sugar. Add melted butter and stir until cohesive, there shouldn't be a slick of butter on the top of the mixture anymore.
Add egg and egg yolk and whisk until smooth; you shouldn't see any streaks of unmixed egg white. Mix in vanilla.
Whisk together flour, milk powder, baking soda, cream of tartar and salt in a bowl until evenly distributed.
Add dry ingredients to wet ingredients and fold until incorporated and no streaks of flour remain. The dough will be quite soft, but not sticky. Refrigerate for 15 minutes to allow it to firm up enough so it will hold a ball shape without slumping.
Use a medium cookie scoop to scoop dough into 1 1/2-inch balls (approximately 40-42g of dough each), then roll in reserved sugar-tea mixture to coat.
Transfer to prepared cookie sheets. Repeat with remaining dough, leaving 2-3 inches of space between cookies to allow for spread.
Bake for 12-14 minutes or until tops are puffed and edges are just starting to brown. Give the sheet a good rap on the oven rack before you take it out of the oven to knock out any air (this produces more evenly flat cookies with the ripples around the edges). You can also swirl a round cookie cutter around the outer edges of the still-warm cookies to tighten up the shapes.
Let cool on baking sheet for 10 minutes, then transfer to a wire rack to cool completely.
For glaze, whisk together sifted powdered sugar with reserved 1/4 teaspoon of tea powder. Add 2 teaspoons of milk and whisk until smooth. Add additional milk, 1/4 teaspoon at a time, until desired consistency is achieved (when drizzled, it should soften back into itself within 3-4 seconds). You can drizzle over cookies with a fork or whisk, or transfer to a piping bag or squeeze bottle for more precise decoration. If desired, sprinkle with a few lavender buds before the glaze sets (but go easy on this, maybe 4-5 buds max per cookie, lest your cookies start tasting like potpourri).
Cookies are best enjoyed within a day or two, but will keep in an airtight container in the refrigerator for up to 5 days, or sealed in a bag in the freezer for up to 2 months.