Preheat the oven to 350℉.
In a bowl, melt the butter and allow it to cool a bit. Next, add in the eggs, sugar, and the vanilla extract. Then, zest the 2 lemon into the wet ingredients and mix everything until incorporated.
In another bowl, mix together the flour, salt and baking powder. Then, combine the dry and wet ingredients.
Pour the batter into the baking dish, then place into the oven. Bake for 20-25 minutes. Allowing them to undercook slightly will give more of the underbaked brownie feel. The longer they bake the more cake-like they'll be.
Remove from oven and allow them to cool completely.
Before serving, whisk together the ingredients for the icing until smooth then spread over the cooled lemon brownies. Once the icing has set, cut then serve.