Make the crust. Combine graham crumbs, brown sugar and melted butter in a medium bowl and blend until well combined. Press the crumbs firmly and evenly into the bottom and just about ½-inch up the sides of the pan. I like to use a straight edged drinking glass or the bottom of a ½ cup measuring scoop to pack it evenly and smooth it out along the bottom and push it gently up the sides. For a 9-inch pan, you will just have enough to make a flat crust. Bake for 10-12 minutes until lightly golden and feels dry. Meanwhile, make the filling.
Reduce oven temperature to 300°F.
Make the filling. Mix cream cheese in a large bowl with an electric hand mixer on medium speed until smooth. Gradually add sugar and mix until very smooth and creamy. Stop to scrape down the sides of the bowl several times to ensure there are not lumps at this stage and everything is well combined. The mixture should be lump-free before you move on to the next step.
Add water (and lemon juice if you're using it) and mix until smooth, then mix in sour cream, vanilla extract and vanilla bean paste until just combined. Scrape down the sides and around the bottom of the bowl again and mix on low for 5 seconds to ensure everything is blended.
Add one egg and mix on medium speed until JUST combined. Scrape down the sides and bottom of the bowl, add the other egg and mix just until incorporated.
Pour the batter onto the pre-baked crust (it's ok if it is still warm) and spread it out evenly. Tap the pan on the counter top 10 times and pop any air bubbles that you see rising to the surface, then use the pointed end of a knife or skewer to pop any large air bubbles that float up. Place cheesecake in the oven on the center rack. Place a large metal rimmed baking sheet or a roasting pan on the bottom oven rack of the preheated oven. Pour boiling water into the pan about 1 inch deep.
For an 8-inch pan with the smaller batch, bake for 35 minutes. Turn the oven off and, without opening the oven door, leave it in for another 15 minutes so it cooks gently. This will create the silkiest texture. It is ready when the filling is set and slightly puffed around the edges, but the center is slightly wobbly or jiggles when the pan is shaken. The jiggly portion should span no greater than 3 inches through the middle.For a 9-inch pan with the larger batch, bake for 50 minutes. Turn the oven off and, without opening the oven door, leave it in for another 15 minutes.
NOTE: if you're unsure about your oven, you can open the door to check on the cheesecake after the 30 minute mark for an 8-inch pan and 45 minutes for a 9-inch pan. At this point it should look matte and set around the edges about 1 ½ inches inward, and then it should be quite wobbly in the middle. If all looks well, close the door and finish baking at 300°F for the remaining 5 minutes then proceed to turn off the oven and leave it inside the slowly cooling oven for 15 minutes.
Transfer pan to a wire rack to cool. Immediately run a thin knife or small offset spatula around the edges of the cake to release it from the sides and then leave it to cool completely.
Once completely cooled, refrigerate for at least 4 hours before serving. I like to leave mine overnight and serve it the next day.