Preheat oven to 325°F. Line a baking sheet with parchment paper, set aside.
Cream the softened butter, vegetable shortening and sugars until fluffy - about 2 minutes. (You may want to use a mixer for this part)
Add in the egg and vanilla and mix well.
Add the flour, baking powder, baking soda, salt and cayenne pepper. Stir with a wooden spoon until the dry ingredients are about 50% incorporated.
Add the white chocolate morsels and lime zest.
Continue mixing all ingredients until fully incorporated. Cover dough with plastic wrap, place in the refrigerator. Chill dough for 1 hour.
Using a cookie scoop, or soup spoon, drop rounded balls of cookie dough onto the parchment-lined baking sheet, about 2 inches apart. (If using a soup spoon, aim for dough balls that are larger, roughly 1" in diameter).
Bake for 18-21 minutes or until the cookies are just starting to lightly brown around the edges and the centers are set but light.
Allow baked cookies to cool on the sheet pan for about 5 minutes before removing to a wire rack to cool completely.
If desired, combine all glaze ingredients in a small mixing bowl and stir to combine. Brush or drizzle on cooled cookies and allow to set.