Beat the cream cheese and sugar together until smooth. Add the egg and vanilla, and stir to combine.
Preheat the oven to 350°F (176°C). Drain the grated zucchini and squeeze out as much moisture as possible. Set aside.
Lightly grease two 8x4-inch (20 x 10 cm) loaf pans. Melt the butter and let it cool without solidifying, then stir in the sugar. Beat in the eggs, vanilla, and cinnamon, and add the zucchini, stirring to combine well.
In a separate bowl, combine the all-purpose flour, salt, baking soda, and baking powder. Add the milk (or buttermilk) alternately with the flour mixture, stirring to combine; the batter will be lumpy.
Pour enough batter to cover the bottom of the pans. Carefully divide the cheesecake batter between the two loaf pans, spreading it to the edges.
Cover the cheesecake batter with the remaining batter.
Bake for 45–55 minutes, or until a cake tester comes out clean. Allow to cool completely before serving.