Preheat the oven to 350F and line two large baking sheets with parchment paper.
In a small saucepan on medium heat, melt the butter. Transfer to a large mixing bowl to cool, set aside.
In a medium-sized bowl, whisk the flour, baking soda, baking powder, and salt. Set aside.
To the bowl with melted butter, add the sugar. Mix-in with a spatula until it turns pasty. Add the lemon zest and rub it a bit with your fingertips to release the oils. Then mix the 1 egg, 1 egg yolk, lemon juice, and vanilla until smooth.
Pour in the dry ingredients and use a rubber spatula to fold in and combine. Don’t overwork the dough or it will become tough. Fold it together until there are no more traces of flour. Set aside to rest while you preheat the oven. (This allows the dough to hydrate, which I found more important with the addition of acid than the original sprinkle sugar cookies. )
Using a large cookie scoop (OR ¼ cup measuring cup) the cookies onto the baking sheets and bake for 12-14 minutes. Rotate the baking sheets halfway through baking.
The cookies are done when the centers are just set, with a light golden ring around the edges.Remove them from the oven and give the pans a couple of gentle taps on the counter to release any air and help them settle. Then, using the inside of a fork, gently round the edges into shape if needed to create perfectly round cookies.
Let cool for 10 minutes on the pan. Transfer to a cooling rack to cool completely before drizzling with icing. Let the icing harden a bit before eating, it won’t take long!
If freezing, freeze the cookies un-iced and drizzle with icing once thawed and ready to be served.