Step 1: In a large bowl, whisk together flour, 1/4 cup sugar, yeast, and salt.
Step 2: In a separate bowl, whisk warm milk and 1 egg. Pour wet ingredients into dry ingredients and mix until a shaggy dough forms. Add softened butter and knead on a lightly floured surface for 8-10 minutes, until smooth and elastic.
Step 3: Place dough in a lightly oiled bowl, cover, and let rise in a warm place for 1-1.5 hours, or until doubled in size.
Step 4: While dough is rising, prepare the filling: In a bowl, beat softened cream cheese with 1/4 cup sugar until smooth. Beat in egg yolk, vanilla extract, and a pinch of salt until well combined.
Step 5: Once dough has risen, punch it down gently and divide into 12 equal pieces. Roll each piece into a ball.
Step 6: Flatten each dough ball into a disc. Place a spoonful (about 1.5-2 tbsp) of cream cheese filling in the center. Gather the edges of the dough around the filling and pinch to seal, forming a neat bun. Place seam-side down on a parchment-lined baking sheet.
Step 7: Cover the buns loosely with plastic wrap and let them rise for another 30-45 minutes, or until puffy.
Step 8: Preheat oven to 375°F (190°C). Once risen, brush the tops of the buns lightly with an egg wash (1 egg beaten with 1 tbsp milk or water).
Step 9: Bake for 20-25 minutes, or until golden brown and the filling is slightly caramelized at the edges. For a more 'burnt' top like a Basque cheesecake, you can briefly broil for 1-2 minutes, watching carefully to prevent burning.
Step 10: Let the buns cool slightly on a wire rack before serving. Enjoy warm!