Prepare the yeast: Warm milk slightly. Stir in sugar and yeast. Let sit 5–10 minutes until foamy.
Make the dough: In a large bowl, mix flour and salt. Add yeast mixture, melted butter, egg, and vanilla. Knead 8–10 minutes until smooth. Cover and let rise for 1 hour.
Make the filling: Mix flour, sugar, softened butter, and vanilla until crumbly.
Shape the Gata: Punch down dough. Divide into 2–3 portions. Roll each into a circle ~½ inch thick. Spread filling. Roll into a log and shape into a circle or braid. Place on a greased baking tray.
Bake: Preheat oven to 350°F (175°C). Bake 25–30 minutes until golden brown. Let cool slightly.
Serve: Slice and enjoy warm. Dust with powdered sugar if desired.