Preheat the oven to 350°F. Lightly grease 2 baking sheets or line with parchment paper or silpat mats.
MAKE THE COOKIES:
In a medium bowl, use an electric mixer to combine the butter, sugar, vanilla, baking powder, baking soda and salt. Beat in the egg, then the orange zest and juice, scraping the bowl. The mixture will look curdled; that's OK! Add the flour, beating until smooth. Drop the dough by tablespoonfuls (or use a cookie scoop) onto the prepared baking sheets.
Bake the cookies 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.
MAKE THE ICING:
In a medium bowl, use an electric mixer to combine the sugar, butter, orange zest, vanilla and salt until well combined. Beat in as much orange juice as you need until the mixture is spreadable. Spread the icing on the cookies when they're completely cool, using a generous 1 teaspoon icing for each.