Heat oven to 350°F; line sheets with parchment.
Cream softened butter and sugar in large bowl about 3 minutes until light and fluffy.
Beat in eggs individually; add lime juice, zest, and vanilla.
Separately whisk flour, graham crumbs, baking soda, and salt.
Gradually mix dry into wet just until combined.
Drop rounded tablespoons 2 inches apart on sheets.
Bake 10-12 minutes until edges lightly golden; cool on sheet 5 minutes, then transfer to rack.
Whisk powdered sugar and lime juice smooth; drizzle over cooled cookies and set 15 minutes before serving.