Add the butter to a saucepan and place it over medium heat.
Allow the butter to melt completely, then continue cooking until it turns golden brown and fragrant. Remove from the heat and let it cool slightly.
Transfer the browned butter to a mixing bowl. Add the brown sugar and mix until well combined.
In a small bowl, combine the softened cream cheese, lemon juice, and lemon zest.
Add this mixture to the browned butter mixture and whisk until smooth and fully incorporated.
Add the flour, baking soda, and salt. Mix until a soft dough forms.
Stir in additional lemon zest and mix until evenly distributed throughout the dough.
In a small bowl or plate, combine the granulated sugar and lemon zest for the coating.
Scoop portions of dough and roll them into balls. Roll each dough ball in the lemon-sugar mixture.
Place the coated dough balls on a lined baking sheet, leaving space between each cookie.
Bake at 350°F (175°C) for 10–12 minutes, or until the edges are lightly golden.
Allow the cookies to cool completely on a wire rack.
In a small bowl, whisk together the powdered sugar, lemon juice, and lemon zest to make the glaze.
Drizzle the glaze over the cooled cookies and let it set before serving.