Put about a fourth of the parsley and mint in a food processor and blend on low speed. Slowly increase the speed to medium and continue adding the rest of the herbs until you have added all of them. In a steady stream, add half of the olive oil. Mix on medium-low speed (or pulse, if using a food processor) until combined. Add the remaining olive oil and blend for no longer than 10 seconds. The mixture will look loose and stringy. Scrape out the blender to get all of the parsley mixture, transfer to a bowl, and refrigerate until ready to use.

In a bowl, combine the flour, cornstarch, salt, and baking powder; set aside.

In a stand mixer fitted with the paddle attachment, whip the eggs for about 30 seconds. Add the sugar and mix on high speed until the mixture is very thick and turns pale yellow, about 3 minutes. Turn the mixer speed down to low and add the herb-oil mixture. With the mixer still running, add the flour mixture and mix until just combined. Do not over mix. Pour the batter into a container and refrigerate for at least 6 and up to 24 hours. (The cupcakes will turn out much greener than if you bake them right away).

Preheat the oven to 340°F. Grease the cupcake pan and insert liners. Pour the batter into liners, filling them about two-thirds full.

Bake for 22–24 minutes. Insert a toothpick to test for doneness. If the tops begin to brown before the cupcakes are done, turn the heat down to 330°F and let them bake 2–3 minutes longer. When a cake tester inserted in the center of the cupcake comes out clean, they are done. Let cupcakes cool in the pan. Remove before frosting.

Cream cream cheese with an electric mixer. Add butter and cream together until light and fluffy. Add confectioner’s sugar, 1/2 cup at a time. After each 1/2 cup has been incorporated, turn the mixer to the highest speed for about 10 seconds to lighten the frosting. Add vanilla and beat until well blended, light, and fluffy.