Pulse flour, powdered sugar, salt, lemon zest and lavender in a food processor until combined.
With the processor running, add butter one tablespoon at a time until moist and crumbly.
Add vanilla and lemon juice and process just until a dough ball forms.
Gather dry bits, shape the dough into a disk and lightly dust with flour.
Line an 8- or 9-inch square pan with parchment and press the dough into an even layer.
Cut the dough into squares; use 5 by 5 sections for 25 cookies in an 8-inch pan, or 6 by 6 for 36 in a 9-inch pan.
Prick the dough all over with a toothpick or skewer to dock it.
Refrigerate the shortbread for 30 minutes.
Heat the oven to 300 F with a rack in the center.
Bake about 45 minutes, until golden brown and baked through.
Immediately recut the shortbread, then cool completely in the pan before lifting it out.
Stir powdered sugar and meringue powder with lemon juice until smooth and drizzly.
Dip cookie tops into icing, shake off excess, and decorate with lavender or zest before it sets.
Let icing dry and set completely for at least 4 to 6 hours.
Store airtight at room temperature for 1 to 2 weeks or freeze for 3 to 6 months.