Preheat your oven to 350°F (175°C). Thoroughly grease and flour a 10-inch bundt pan. Set it aside.
To make the filling, beat 8 oz of softened cream cheese, the softened butter, and 1/2 cup of granulated sugar with an electric mixer until smooth. Add 1 egg, 2 tablespoons of flour, and 1 teaspoon of vanilla, and beat just until combined. Set this mixture aside.
In a medium bowl, whisk together 2 1/2 cups of flour, 1 3/4 cups of granulated sugar, the cocoa powder, baking soda, and salt.
In a separate large bowl, whisk together the oil, 2 eggs, sour cream, milk, 1 tablespoon of vanilla, and vinegar. Gradually mix the dry ingredients into the wet ingredients until just combined. Stir in the red food coloring until you reach your desired shade.
Pour about half of the red velvet batter into the prepared bundt pan. Spoon the cream cheese filling evenly over the batter, keeping it about an inch away from the edges. Gently cover the filling with the remaining batter.
Insert a butter knife into the batter and swirl it two or three times around the pan to lightly marble the filling into the cake. Do not overmix.
Bake for 55 to 65 minutes. The cake is done when a wooden skewer inserted into the center comes out clean. Let the cake cool in the pan on a wire rack for at least 1 hour before inverting it onto a serving plate. Allow it to cool completely.
For the glaze, beat 4 oz of softened cream cheese, the powdered sugar, and heavy cream until smooth. Drizzle the glaze over the completely cooled cake before slicing and serving.