Preheat oven to 350 degrees F and line a 9x9 square baking pan with parchment paper.
In a food processor, pulse together the flour, sugar, salt and lavender until well blended. You want the lavender to be dispersed and chopped up a bit.
Add in the cubes of softened butter and pulse until the mixture is sandy and starts to form clumps.
Gently press shortbread mixture into an even layer over the bottom of the prepared pan. Use a light hand when pressing in your dough to keep the shortbread crumbly and cookie-like. Refrigerate for a minimum 20-30 minutes.
Bake crust for 22-25 minutes or until the edges start to turn a golden brown.
Remove shortbread from the oven, let it cool slightly and reduce the temperature to 325 degrees F.
Prepare the lemon filling. In a medium bowl, rub together the sugar and lemon zest between your hands to release the oils from the zest. Whisk in the flour until blended. Add the eggs and whisk briskly so that the sugar doesn't "cook" the eggs and form clumps.
Add the lemon juice and half and half and whisk to fully incorporate.
Pour filling over the warm shortbread and bake for an additional 25-30 minutes, until the center jiggles very (very) slightly.
Let the bars cool to room temperature, about an hour. The longer they cool, the easier and prettier they are to slice. Transfer bars to a cutting board and cut into squares or wedges. Using a sieve, sprinkle with powdered sugar and serve.