Activate the Yeast: In a stand mixing bowl, combine the warm milk (105-115°F) and sugar. Sprinkle the yeast over the milk mixture and stir gently. Let it sit for about 5-10 minutes, until the mixture is frothy. This ensures the yeast is activated.
Once yeast is activated, add flour, salt, vanilla extract, and eggs to the stand mixing bowl. Using the dough hook attachment, knead on low speed for 2-3 minutes until mixture comes together. Once the dough is formed, increase to medium speed and continue to knead for 5-7 minutes until the dough is smooth and elastic.
Gradually add the room temperature butter in two parts, allowing the butter to be fully incorporated in the dough before adding the second part. Continue kneading until the dough is fully incorporated and the butter is fully absorbed, about 8-10 minutes. The dough should be soft, slightly sticky, but should pull away from the sides of the bowl.
Once the dough is smooth and elastic, remove it from the bowl and place it in a lightly greased bowl. Cover with plastic wrap, and let the dough rise in a warm environment for about 1 hour, or until doubled in size.
Once the dough has doubled in size, gently punch it down to release the air. Turn the dough out onto a lightly floured surface. Divide the dough into 12 even pieces. To do this, divide the dough in half, then divide each half into 6 even pieces (you can use a scale for more accurate measurement or just eyeball it). For each piece, gently fold the edges into the center, forming a tight seam. Roll the dough into a ball, ensuring the seam is on the bottom and the surface is smooth and tight. Arrange a large cookie sheet with 12 parchment paper squares and place the shaped brioche dough on parchment paper, place a piece of plastic wrap onto of dough. Allow to rise at room temperature for 30 minutes, or until puffed up.
Heat oil in a deep fryer or large pot to 340-350°F, using a candy thermometer to monitor the temperature. Carefully drop the risen dough balls with parchment paper into the hot oil, a few at a time (I did 3 at a time). Once dropped in oil, remove parchment paper out from under the donut using tongs. Fry for about 2-3 minutes on one side, then flip the donuts and fry for another 2-3 minutes on the other side, or until golden brown and cooked through. Remove the donuts from the oil and place them on a plate lined with paper towels to drain excess oil. Let donuts cool while making the Chocolate Chip Cookie Dough Filling.