Preheat your oven to 350°F (180°C). Line the bottom of a 9-inch cake pan with parchment paper and lightly grease the sides with baking spray.
In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
In a large mixing bowl, beat the butter and sugar on medium speed for 2 to 3 minutes, until light and fluffy.
Add the egg and vanilla extract, then mix until fully combined.
Mix in the lemon juice and lemon zest. The mixture may look slightly curdled, which is normal.
Add the dry lemon pudding mix, then add the flour mixture. Mix just until combined. Do not overmix the dough.
Fold in the white chocolate chips.
Press the dough evenly into the prepared pan. Add a few extra white chocolate chips on top if desired.
Bake the Lemon Cookie Cake for 19 to 24 minutes, or until the edges are lightly golden.
Let the cake cool completely in the pan before transferring it to a serving plate.
To make the frosting, beat the cream cheese and butter together until smooth.
Add 1 cup of powdered sugar slowly and mix until combined.
Mix in the vanilla extract and lemon zest.
Add the remaining powdered sugar and beat until smooth.
Pipe or spread the frosting around the edges of the cooled cake. Refrigerate until ready to serve, then bring to room temperature before serving.