Heat the oven to 350°F and line a 12-cup muffin pan with cupcake liners.
Whisk together the flour, the baking powder, ½ teaspoon of the salt, and the baking soda in a medium bowl.
Stir together the milk and 1 tablespoon of the lemon juice in a separate small bowl.
Beat the brown sugar and ½ cup of the butter with the paddle attachment of a stand mixer (or with a handheld mixer) until light and fluffy.
Add the eggs, 1 teaspoon of the vanilla extract, 1 teaspoon of the rose water, and 1 teaspoon of the lemon zest, and beat until well combined.
Gently stir in about ⅓ of the flour mixture, followed by about half of the milk mixture. Repeat, and then add the last of the flour mixture and stir just until combined.
Divide the batter among the muffin cups, and bake until the cupcakes are golden brown, 20 to 25 minutes.
Cool while you prepare the frosting.
Beat the remaining ½ cup butter and the powdered sugar with the paddle attachment of a stand mixer (or with a handheld mixer) until well combined.
Add the remaining 2 tablespoons lemon juice, 1 teaspoon vanilla extract, 1 teaspoon rose water, and 1 teaspoon lemon zest along with a pinch of salt. Beat for at least 10 minutes.
Remove the cupcakes from the pan.
Spread the frosting on the cupcakes and serve.
