Combine cream cheese and sugar in the bowl of a stand-mixer and cream together until smooth. Add eggs, one at a time, fully incorporating each before adding the next. Make sure to scrape down the bowl in between each egg. Add heavy cream, vanilla, lemon juice, extracts, and food color and blend until smooth and creamy.
Pour batter into prepared crust and tap the pan on the counter a few times to bring all air bubbles to the surface.
If cheesecake pan is not airtight, cover bottom securely with foil before adding water. Place pan into a larger pan and pour boiling water into the larger pan until halfway up the side of the cheesecake pan.
Bake 45 to 55 minutes, until it is almost done - this can be hard to judge, but you're looking for the cake to hold together, but still have a lot of jiggle to it in the center. You don't want it to be completely firm at this stage. Close the oven door, turn the heat off, and let rest in the cooling oven for one hour. This lets the cake finish cooking and cool down gently enough so that it won't crack on the top. After one hour, remove cheesecake from oven and lift carefully out of water bath. Let it finish cooling on the counter.