Using a stand mixer with a paddle attachment, cream room temperature butter until light and fluffy.
Using a small food processor, pulse sugar, lavender buds, and orange zest until incorporated. Add to butter and mix on medium speed until combined.
Add flour and salt and mix first on low, then on medium until incorporated. Scrape sides of the bowl with a spatula as needed, then add orange juice. Mix until the dough comes together.
Pour mixture over saran wrap. Fold plastic wrap over the dough, as you shape it into a disk. Chill cookie dough in the fridge for 30 minutes.
Using a floured surface and floured rolling pin, roll chilled dough until it's ¼" thick. Use a 2 to 2½ inch cookie cutter to cut out cookies. Press dough scraps together and roll out until all the dough is used up.
Add cookies to a parchment paper or silicone mat-lined baking sheet and chill in the freezer for 15 minutes. Preheat convection oven to 325℉.
Bake chilled cookies in preheated oven for 13-15 minutes or until golden around the edges. Transfer to a cooling rack.
Optional Step
Whisk powdered sugar (icing sugar), lavender syrup, and half and half in a small bowl until smooth in consistency. Drizzle over cooled cookies and sprinkle with lavender buds. If you want to accentuate the orange flavor, use orange syrup and sprinkle with orange zest instead!