Preheat oven to 350°F (175°C) and grease and flour two 9-inch cake pans.

In a medium bowl, whisk together flour, baking powder, baking soda, salt, and lavender; set aside.

In a separate bowl, cream the butter and sugar until pale and fluffy, about 3-4 minutes.

Add eggs one at a time, beating well after each addition.

Add milk, lemon zest, ½ cup oil, lemon juice, and vanilla. Mix until smooth.

Fold in dry ingredients in two additions, mixing just until combined.

Divide the batter between the prepared pans and bake for 30-35 minutes, rotating halfway through.

Check with a toothpick; it should come out clean or with a few moist crumbs.

Cool in pans for 10 minutes, then transfer to a rack to cool completely.

In a small saucepan, warm milk and lemon extract over low heat; do not boil.

In a clean stand mixer bowl, beat the egg whites to soft peaks. Gradually add powdered sugar and beat to stiff peaks.

With the mixer running low, slowly stream in the warm milk mixture, then beat in sour cream and the remaining butter until smooth.

Optionally, fold in finely ground lavender.

Spread frosting on one layer, stack, then crumb-coat and finish-cover the cake.

Decorate with fresh lavender sprigs and thin lemon slices.

Let the cake rest for 30 minutes before slicing.