In a saucepan, heat 2 cups of whole milk over medium heat until it just begins to simmer. Meanwhile, whisk together 4 egg yolks, 1/2 cup granulated sugar, 1/4 cup cornstarch, and a pinch of salt in a separate bowl. Gradually add the heated milk to the egg mixture while whisking continuously. Return to the saucepan, cooking over medium heat until thickened, about 5–7 minutes. Remove from heat, stir in 1 tablespoon of vanilla bean paste, and chill in the refrigerator until completely cold.

In a large mixing bowl, combine 1/2 cup lukewarm milk, 2 tablespoons instant yeast, and 2 tablespoons granulated sugar. Allow it to sit for about 5 minutes until frothy. Add 4 cups of all-purpose flour, 2 eggs, and 1 teaspoon of salt, mixing until it forms a shaggy dough. Gradually incorporate 1/2 cup of cubed unsalted butter, kneading for about 10 minutes until the dough is smooth and elastic.

Shape the brioche dough into a ball and place it in a lightly greased bowl. Cover it with a kitchen towel or plastic wrap, and let it rise in a warm area until it doubles in size, approximately 1 to 2 hours. Once risen, gently punch the dough down to release the air and refrigerate it for at least 4 hours, or overnight.

In a saucepan, add 2 cups of fresh blackberries, 1/4 cup granulated sugar, 1 tablespoon lemon juice, and a pinch of salt. Cook over medium heat for about 10 minutes, stirring gently until the blackberries break down and release their juices. In a small bowl, mix 1 tablespoon of cornstarch with 2 tablespoons of water and stir into the blackberry mixture. Cook until thickened, about 2–3 more minutes, then remove from heat and let cool completely.

On a lightly floured surface, divide the chilled brioche dough into 12 equal pieces. Roll each piece into a ball, then flatten slightly. Place them on a lined baking sheet, ensuring they are evenly spaced. Cover the buns with a clean towel and allow them to proof for about 1.5 to 2 hours, or until they are puffy and have doubled in size.

Preheat your oven to 375°F (190°C). Once the buns have proofed, use your finger to create a small indentation in the center of each bun. Brush the edges with a beaten egg wash to help seal. Pipe in a generous amount of chilled vanilla pastry cream, followed by a dollop of blackberry filling, being careful not to overfill.

Place the baking sheet in the preheated oven and bake for 20 minutes, or until the brioche buns are golden brown on top. Once baked, remove them from the oven and immediately brush with a sugar syrup made from 1/4 cup granulated sugar dissolved in 1/4 cup hot water for a glossy finish. Allow to cool slightly before serving.