In a stand mixer fitted with the paddle attachment, cream the butter on medium speed until light and fluffy, then add the shortening and beat until evenly mixed in. Add the granulated and brown sugar and beat on medium speed for 3 minutes to combine. Mix in the vanilla. Add the eggs one at a time and mix on medium-low speed just until incorporated.
In a medium bowl, whisk together the flour, salt, and baking soda. Add the flour mixture to the butter mixture and mix just until incorporated. Mix in the chocolate.
Scoop the cookie dough onto a parchment-lined baking sheet using a portion scoop (I used a 3 ounce/83g scoop). You can make the cookies larger or smaller, but it will affect the baking time.
Refrigerate the dough for at least 30 minutes if you are in a yank, but the cookies will improve if you let them sit for 24 to 36 hours before being baked. Resting will make them taste better, be more uniform in shape and color nicely when they bake. After they are chilled you can bake them or freeze the dough balls for later baking.
Position a rack in the middle of the oven and preheat the oven to 375°F.
Evenly space 6 chilled cookie balls on a parchment-lined baking sheet and sprinkle with sea salt. Bake until set in the middle and golden brown around the edges, 12 to 15 minutes. Allow the cookies to cool slightly on the pan and then remove to a wire rack. Repeat with the remaining dough balls.