Preheat oven to 350 degrees F (175 degrees C).
Fill a large oven-safe pot with water to 1 inch; bring to a boil.
Place sweetened condensed milk, evaporated milk, eggs, cream cheese, milk, and vanilla extract in a blender; blend until smooth.
Place sugar in an even layer in a saucepan over medium heat. Cook until edges start to brown, about 1 minute; drag sugar into the center with a spatula once the edges start to brown. Continue cooking, stirring from time to time, until caramel is an even golden brown, 4 to 5 minutes.
Pour caramel into a 9-inch flan mold or baking pan; swirl so that caramel reaches 1 inch up the sides. Pour the condensed milk mixture on top. Place mold in the pot of boiling water.
Bake flan in the preheated oven until a damp table knife inserted into the center comes out clean, 45 minutes to 1 hour. Remove from oven and let stand for 5 minutes.
Chill flan in the refrigerator until firm, 2 to 3 hours. Invert onto a serving plate so that caramel is on top.