Begin by making the graham cracker crust. Preheat the oven to 350F. Add the graham crackers to a food processor or blender. Blend until crumbs remain. Add the melted butter, brown sugar, and salt. Blend again until everything is combined.
Line a 10 inch springform pan with parchment paper (on the bottom and around the edges). Pour the graham cracker mixture into the pan then use your fingers to spread it out, then to pack it on the bottom and up the edges, forming a crust. Transfer to the oven for 10 minutes and bake until its just starting to brown. Set aside to cool while you make the filling.
Before making the cheesecake layer, turn the oven down to 325F. Then, fill a 9x13" pan with water and put it on the bottom rack of your oven. You will leave this pan in the oven the entire time the cheesecake bakes. It will create a steam bath that helps the cheesecake bake evenly, and prevents cracks from developing.
Now make the cheesecake layer, add the sugar and lemon zest to a bowl. Use your hands to rub the sugar and lemon zest together between your fingers. This will release the lemon oils and make the lemon flavor stronger. Add the lemon sugar to the bowl of a stand mixer (a hand mixer will also work). Add the room temp cream cheese and mix for 3-4 minutes, until totally smooth.
Scrape down the sides of the bowl then add the lemon juice, plain yogurt, and vanilla extract. Mix for another 2-3 minutes. Scrape down the sides of the bowl again.
Add the eggs, one at a time, mixing until just combined each time. Don't over mix here! Once the last egg has been added, stop mixing once you can't see streaks of egg anymore.
Pour the cheesecake mixture into the graham cracker crust (it is fine if the crust is still warm) and level it out with a spatula.
Transfer the cheesecake to the oven and bake for 60-80 minutes, until the cheesecake is still slightly jiggly in the center, but mostly set.
When the cheesecake is finished baking, don't remove it from the oven! Leave it inside. Turn the oven off, prop the oven door open slightly, and leave the cheesecake inside for 1 hour. This will help the cheesecake slowly cool down (which helps prevent cracking).
After that hour, take the cheesecake out of the oven. Leave it on the counter to cool for another hour. After that, cover the cheesecake and transfer it to the fridge for at least 4 hours or overnight to cool completely.
Just before serving, top the cheesecake with lemon zest. Slice, serve, and enjoy!!