Preheat oven to 350 degrees F. Lightly grease two baking sheets or line with parchment paper.
Cream the 11 tablespoons butter, sugar, 1 teaspoon vanilla, baking powder, baking soda and ½ teaspoon salt in a medium bowl.
Beat in the egg, then the ½ cup orange juice and 2 tablespoons orange zest, scraping the bowl. The mixture will look curdled—that's OK.
Add the flour, beating until smooth.
Drop the dough by tablespoonfuls onto the prepared baking sheets.
Bake the cookies, reversing the pans midway through (top to bottom, bottom to top), until they're just barely beginning to brown around the edges, 10 minutes.
Remove the cookies from the oven and let them cool on the pans for 10 minutes before transferring them to a rack to cool.
To prepare the icing: Beat the confectioners' sugar, 2 tablespoons butter, 1 tablespoon orange zest, ¼ teaspoon vanilla and salt in a medium bowl until well combined.
Beat in the 2 tablespoons orange juice till the mixture is spreadable.
Spread icing on the cookies when they're completely cool, using a generous 1 teaspoon of icing for each.