Step 1: Prepare the Dough

1. In a small bowl, combine warm water, warm milk, sugar, and yeast. Stir gently and let it sit for 5–10 minutes until frothy.
2. In a large mixing bowl, combine flour and salt.
3. Add the yeast mixture, egg, and softened butter to the flour. Mix until a dough forms.
4. Knead the dough on a lightly floured surface for 8–10 minutes until it’s smooth and elastic.
5. Place the dough in a greased bowl, cover with a damp towel, and let it rise in a warm spot for 1–2 hours or until it doubles in size.

Step 2: Make the Cheesecake Filling

1. In a medium bowl, beat the cream cheese and sugar together until smooth.
2. Add the vanilla extract and egg yolk. Mix until fully combined. Cover and refrigerate until ready to use.

Step 3: Prepare the Streusel Topping

1. In a small bowl, mix flour and sugar.
2. Add the cubed butter and use your fingers or a pastry cutter to combine until the mixture resembles coarse crumbs. Set aside.

Step 4: Shape the Buns

1. Punch down the risen dough and divide it into 8–10 equal portions. Shape each piece into a ball.
2. Place the balls on a baking sheet lined with parchment paper, spacing them evenly.
3. Use the back of a glass or your fingers to press a deep well into the center of each dough ball, creating a bun shape.

Step 5: Fill the Buns

1. Spoon a generous amount of the cream cheese filling into each well.
2. Top the cream cheese filling with a small spoonful of strawberry jam and fresh strawberries (if using).
3. Sprinkle the streusel topping over the buns.

Step 6: Final Rise and Bake

1. Cover the buns lightly with a towel and let them rise for 20–30 minutes.
2. Preheat your oven to 375°F (190°C).
3. Brush the edges of the buns with egg wash for a golden finish.
4. Bake for 18–22 minutes or until the buns are golden brown and the filling is set.

Step 7: Cool and Serve

1. Allow the buns to cool on the baking sheet for 10 minutes before transferring them to a wire rack.
2. Serve warm or at room temperature.