FOR THE COOKIES: Preheat oven to 350 degrees. In a food processor or blender, combine greek yogurt, milk, dried lavender and vanilla and blend for 1 minute, scraping down sides and lid of food processor once during blending.
Pour mixture through a fine mesh strainer into a bowl while using a spatula to press and rub lavender against strainer to release oils (also scrape out any excess from food processor - get all of it). Set aside and allow to rest.
In a mixing bowl, whisk together flour, baking powder, baking soda and salt for 30 seconds. In a separate mixing bowl, using an electric hand mixer whip together butter, oil and sugar on medium-high speed until pale and fluffy, about 2 minutes.
Blend in eggs one at a time, adding in food coloring with second egg if using (I used about 30 drops blue and a little less red). Add in flour mixture in three separate batches alternating with half of the milk mixture, beginning and ending with flour mixture and mixing on low-speed just until combined after each addition.
Transfer batter into a piping bag fitted with a 1/4-inch or 1/2-inch round tip and pip cookies onto a Silpat or parchment paper lined baking sheet into 1 3/4-inch rounds, spacing cookies 2-inches apart.
Bake in preheated oven about 6 - 8 minutes. Remove from oven and immediately transfer cookies to a cooling rack. Allow to cool completely then spread frosting on bottom of one cookie and sandwich to bottom of another cookie. Store in an airtight container.
FOR THE FROSTING: In the bowl of an electric stand mixer fitted with the paddle attachment, whip butter and vanilla bean seeds on medium-high speed until nearly white and very fluffy, about 6 - 7 minutes, frequently scrapping down the sides and bottom of the bowl.
Add in powdered sugar, milk, heavy cream and vanilla extract and mix on low speed until blended, then increase speed to medium and beat until very light and fluffy, about 5 - 6 minutes, frequently scraping down the sides and bottom of the bowl.