1. Preheat the oven to 350°F and grease a 9×5-inch loaf pan. Line it with parchment paper.

2. In a medium bowl, beat the cream cheese, 1/4 cup granulated sugar, egg yolk, and 1/2 teaspoon vanilla extract until smooth. Set aside.

3. In another bowl, whisk together the flour, baking powder, baking soda, and salt.

4. In a large bowl, beat the butter and 3/4 cup granulated sugar until light and fluffy.

5. Add the eggs one at a time, then mix in the sour cream, milk, 2 tablespoons lemon juice, lemon zest, and 1 teaspoon vanilla extract.

6. Add the dry ingredients to the wet ingredients and stir just until combined.

7. Spread half the batter into the loaf pan. Spoon the cream cheese filling over the center, leaving a small border around the edges. Top with the remaining batter.

8. Bake for 50 to 60 minutes, or until a toothpick inserted into the bread portion comes out mostly clean.

9. Cool in the pan for 15 minutes, then transfer to a wire rack to cool completely.

10. Whisk together the powdered sugar and 1 to 2 tablespoons lemon juice, adding 1 teaspoon milk if needed. Drizzle over the cooled bread, then slice and serve.