Preheat oven to 350 F (200 C). Grease and flour a 11x5 loaf pan.
In the bowl of a stand mixer beat eggs and sugar for 2 minutes, then add vanilla and oil and mix until combined.
Add baking soda to greek yogurt, stir and add to egg mixture.
Add shredded zucchini to mixture and stir until evenly distributed.
In a separate bowl combine flour, salt, cocoa powder, baking powder and chocolate chips.
Add dry ingredients to wet ingredients and mix until just combined.
Cheesecake layer: In the bowl of a stand mixer with paddle attachment mix cream cheese, sugar, flour, vanilla and lemon zest. Add egg and beat until evenly distributed. Make sure you scrape sides of the bowl, to achieve a homogenous mixture.
Pour 1/2 of zucchini bread batter into baking pan. Carefully spread cream cheese mixture over it. Top with the remaining zucchini bread batter.
Bake for 1 hour and 20-30 minutes, until toothpick inserted comes out clean. Turn off the oven, but leave bread inside for 5 more minutes. Then take out of the oven and let it sit in the pan for 25 more minutes. Transfer to a cooling rack until it cools down completely.
Baking cream cheese filled breads can be tricky. Please make sure you watch your bread and allow extra time to bake. Baking times vary depending on the kind and size of baking pan used, oven type, climate, temperature of the ingredients.