Beat the butter and 3/4 cup powdered sugar together until smooth.
Add 1 tablespoon of lemon juice and vanilla to the mixture, mixing again until combined.
In a separate bowl, soft the flour, salt, lemon zest, and thyme leaves. Slowly add this to the buttercream until just combined.
Roll the dough into a ball, cover in plastic wrap and refrigerate until firm, about 90 minutes.
Meanwhile, preheat the oven to 325ºF.
Line baking sheets with parchment paper.
Roll dough onto a lightly floured surface to 1/4″ thickness. Use cooking cutters to cut the dough into desired shapes, and place onto a baking sheet.
Bake for 10–12 minutes, then allow to cool.
To make icing, combine the remaining 1 cup of powdered sugar with remaining 2 tbsp lemon juice. Ice and garnish with extra thyme if desired.