In a large mixing bowl, combine the warm milk, instant dry yeast, and granulated sugar. Whisk the mixture thoroughly and let it rest for 15–20 minutes, allowing it to become frothy and foamy.
Gradually add the remaining warm milk to the yeast mixture. While continuously whisking, slowly pour in the melted vegetable shortening. Incorporate both the granulated sugar and light brown sugar, stirring until well blended.
Stir in the whisked eggs, anise seeds, and vanilla extract, mixing until the ingredients form a smooth, homogeneous mixture.
Add the all-purpose flour and salt to the bowl. Use a wooden spoon to combine the ingredients until they form a loose dough.
Transfer the dough to a lightly floured surface. Knead for 12–16 minutes, pushing and folding the dough until it becomes smooth and elastic. Shape the dough into a ball.
Lightly grease a large bowl with cooking spray. Place the dough ball in the bowl, cover it with plastic wrap or a clean kitchen towel, and set it in a warm area to rise. Let it rest for 1–3 hours, or until it doubles in size.
Punch down the risen dough on a floured surface to release air. Divide the dough into 10–12 equal portions and roll each into a small ball.
Preheat your oven to 390°F (200°C). Line two baking trays with parchment paper or silicone baking mats. Arrange 5–6 dough balls on each tray, leaving approximately 5 inches of space between them. Gently press each ball with your hands to slightly flatten them.
Cover the trays with kitchen towels and allow the buns to rise again for 15–20 minutes. Lightly dust the tops with flour.
Bake one tray at a time on the middle rack for 10–15 minutes, or until the buns are golden on the top and bottom. Remove from the o