Preheat oven to 350 degrees.
Spray a glass 9-inch square baking pan with cooking spray and set aside.
To get 3/4 cup of pretzel pieces you'll need to crush about 1 1/2 cups of pretzels.
Mix together the pretzel pieces, 1/4 cup of sugar and 4 tablespoons of melted butter.
Spread evenly in the bottom of the prepared baking dish.
With a hand or stand mixer, beat together the remaining 4 tablespoons of softened butter and 3/4 cup of sugar until pale and fluffy, high speed, 2 - 3 minutes.
Add the tablespoon of lime zest and juice, the vanilla, the egg and egg white and beat until smooth.
In a separate bowl, whisk together the flour, baking powder, baking soda and salt.
Add half the flour mixture to the batter and mix on low until combined. Add the buttermilk and mix on low until combined. And then add the remaining flour mixture to the batter and mix on low until a smooth batter is formed.
Spread the batter evenly on top of the pretzel crust.
Bake the cake until it just springs back gently in the center and/or a toothpick inserted into the center of the cake comes out clean, about 25 - 30 minutes.
Allow the cake to cool completely in the pan.
When ready to serve the cake, make the whipped cream.
With a hand or stand mixer, beat together the whipping cream, remaining lime zest and juice and powdered sugar on high until stiff peaks are beginning to form.
Top the cake with the whipped cream and fresh lime wedges.
Slice the cake and serve.