Place your Garlic (2 clove) in the food processor; pulse until finely chopped.
Add the Pistachios (1/3 cup), the juice of 1 Lemon (1), Fresh Basil (4 cup), Salt (to taste), and Ground Black Pepper (to taste); pulse until nuts are finely chopped and ingredients are well blended.
Add Extra-Virgin Olive Oil (1/2 cup) gradually and process until the mixture is well combined.
Add Parmesan Cheese (1/2 cup) and pulse 2 or 3 times. Voila! Pistachio Pesto. Set aside.
In a small bowl combine the Active Dry Yeast (1/2 package), Whole Milk (1/4 cup), and Granulated Sugar (2 1/2 tablespoon) together allowing it to rest until the yeast has been activated and appears foamy, approximately 5 minutes.
Using a large bowl, hand whisk the All-Purpose Flour (2 cup), Fine Sea Salt (3/4 teaspoon), and Vanilla Extract (1/2 teaspoon) to combine. Attach bowl to kitchen aid with the dough hook and add yeasted milk combination to combine.
Add Egg (2) one at a time, mixing thoroughly for a total of 3 minutes. If the dough is still sticking to the side of the bowl and quite wet, you may need to add an extra tablespoon of flour.
Once the dough is not sticking to the side of the bowl anymore, it’s time to add the softened butter in two separate parts. Add the first half of your Unsalted Butter (2 1/2 tablespoon) and mix for 5 minutes.
Scrape down the side of the bowl, then add the last half of the Unsalted Butter (2 1/2 tablespoon) and mix for an additional 6 minutes. As before, if the dough is still sticking to the side of the bowl is sticking, you may need to add an extra tablespoon of flour.
Place the dough into a large clean bowl that is lightly coated in Oil (as needed). Cover in plastic wrap and let is slowly rise for 1-2 hours. Degas the dough by lightly pressing with your hands and place in the fridge overnight so that the flavor has plenty of time to develop.
Flour your counter and roll out the dough until it is about 9x15 inch and spread your pistachio pesto evenly over the dough. Roll up the rectangle so that the rolled log is 15-inches long and place in the freezer for 10 minutes. This will help to cleanly cut through the babka and allow for easy twisting.
Once chilled, cut the log in half with a serrated knife so that you have two 15-inch pieces and the filling is exposed.
Fold the two separate logs on top of each other keeping the sides up as you do so, creating a long twist of pesto-filled dough!
Place the twist in a greased loaf pan. Cover and place in a warm area for 1 1/2 hours until the dough has slightly risen and feels puffy.
Preheat your oven to 350 degrees F (180 degrees C).
Bake your babka for about 35-45 minutes until the top is a golden brown shade. If you happen to have a thermometer handy, check that the interior of the babka is 185-200 degrees F (85-95 degrees C) in the thickest center part of the bread and remove it from the oven.
Transfer to a cooling rack before serving and enjoy!