Start by gathering all your ingredients. Preheat your oven to 350°F (175°C). Line muffin tin with parchment paper or muffin liners.
In a large mixing bowl, whisk together 3/4 cup of granulated sugar, vegetable oil, applesauce, eggs, and sour milk (or buttermilk) until smooth.
In another bowl, mix the all-purpose flour, baking soda, cinnamon, and salt. Stir until evenly distributed.
Gradually add the dry mixture to the wet mixture, stirring gently with a spatula until just combined.
Gently fold in the shredded zucchini until evenly incorporated into the batter.
In a small bowl, mix the softened cream cheese with the remaining 6 tablespoons of sugar until smooth.
Spoon a small amount of the muffin batter into each muffin cup, add a dollop of cream cheese mixture, and top with more batter to cover.
Bake for 20-25 minutes or until a toothpick inserted into the center comes out clean.
Allow muffins to cool in the tin for about 5 minutes before transferring to a wire rack to cool completely.
