Preheat the oven to 350 °F and place parchment paper on two cookie sheets.
In a big bowl, beat butter, shortening, brown sugar, sugar, eggs, honey, and vanilla until it becomes fluffy.
In another large bowl, combine flour, cornmeal, cornstarch, salt, baking powder, and lavender.
Add the dry ingredients to the bowl with the butter mixture, and mix until you get a thick batter.
Roll the dough into small 1-inch balls and put the dough balls on the prepared cookie sheets and press them down to make flat discs.
Bake the cookies for about 15-18 minutes, or until the edges turn slightly brown.
Let the cookies cool on a wire rack.