Use a mortar and pestle to gently grind lavender, lemon zest and granulated sugar together to release some of the oils, or finely chop dried lavender and lemon zest together with a knife before blending with sugar.
Combine flours, confectioner’s sugar, lavender-lemon sugar, and salt in the bowl of a food processor. Pulse to combine.
Add cold butter all at once and pulse until the dough begins to come together but is still slightly crumbly. It is important not to overwork the dough. Gather into two disks and place in a lidded container to chill for at least 2 hours.
Take the dough out of the refrigerator and let it come to a still slightly cool but workable temperature.
Preheat the oven to 325 degrees.
Roll out one disk at a time between two sheets of parchment to 1/8th of an inch thickness.
Use a round or scalloped edge 2-inch cutter (or smaller) to cut out as many cookies as possible, and gather scraps to reroll, chilling as necessary.
Place on a parchment-lined baking sheet. Designate half of the cookies for the tops of the sandwiches, and use a ½ – 1-inch pastry tip to cut out circles in the middle for the jam to peek through; add cutout circles to scrap pile.
Reroll and cut chilled scraps until all of the dough is used. Prick the uncut cookies in the middle with a fork so they don’t puff as they bake.
Bake for 12-14 minutes on the oven’s middle rack until the edges are just golden. Let cool completely.
To assemble, spread a teaspoonful of blueberry jam or preserves to cover each cooled bottom half cookie, and top with each cutout cookie to create sandwiches. If using blueberry jelly, heat until liquid, then cool to room temperature before spreading. Sprinkle with powdered sugar before serving. Store in an airtight container for up to 5 days.