Preheat oven to 350 degrees. Line a 9 x 13 inch cake pan with parchment.
Grate the zucchinis into a bowl. Squeeze excess moisture with paper towels or a clean tea towel - you may need to do this more than once to really get all the water out of the pieces. You should end up with about 1 1/2 cups grated, squeezed zucchini.
In a large bowl, beat the oil and sugars until well combined. Beat in the eggs, one at a time, then the vanilla and Greek yogurt. Add the flour, baking powder, baking soda, cinnamon, and salt. Do not overmix.
Toss the chocolate chips with 2 tablespoons of flour. Fold into the cake batter along with the zucchini.
Pour the batter into the prepared cake pan and tap a couple times on the counter to release any air bubbles. Bake for 30-35 minutes until an inserted toothpick emerges with just a few moist crumbs. Set the tray on a wire rack and let the cake cool completely.
When the cake is cool (sunglasses and all), make the frosting. In a large bowl, beat the cream cheese and butter on medium speed until fluffy. Add in the powdered sugar, vanilla, and instant coffee. Start on low speed to incorporate the powdered sugar, then switch to high until the frosting is airy and soft, about 2-3 minutes.
Lift the cake out of the pan by the edges of the parchment. Frost with the buttercream (or pipe patterns and shit, have fun with it). Slice and serve, or cover and refrigerate for up to 3 days.