Preheat the oven to 350°F (175°C). Line a 12-cup muffin tin with paper liners.
In a large bowl, whisk together the flour, cocoa powder, 1 cup sugar, baking powder, baking soda, and salt until well combined.
In a separate medium bowl, whisk together the milk, vegetable oil, eggs, and 1 teaspoon vanilla extract until smooth.
Pour the wet ingredients into the dry ingredients and stir gently until just combined. Do not overmix.
In a medium bowl, beat the softened cream cheese, 1/3 cup sugar, egg yolk, and 1 teaspoon vanilla extract with an electric mixer until smooth and creamy.
Fill each muffin cup about halfway with the chocolate batter.
Add about 1 tablespoon of the cream cheese mixture on top of the chocolate batter in each cup.
Cover the cream cheese layer with the remaining chocolate batter, filling each cup nearly to the top.
Use a toothpick or knife to gently swirl the cream cheese and chocolate batter together for a marbled effect.
Bake for 18 to 20 minutes, or until a toothpick inserted into the muffin (avoiding the cream cheese center) comes out clean.
Remove the muffins from the oven and let them cool in the tin for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.