For the crust: preheat oven to 325 degrees F. Crush Oreos into fine crumbs (I use my food processor). Combine with the melted butter and press into a foil lined 8x8 pan. Bake for 10 minutes. Remove from oven and make the cheesecake filling.
In a stand mixer, cream the cream cheese and sour cream together until smooth and well combined. Add eggs, one at a time, until combined. Add lemon zest, lemon juice, sugar and vanilla bean paste and mix until just combined.
Add 3 tablespoons of the cheesecake mixture to 6 separate small bowls. Color each bowl with the food coloring to make red, orange, yellow, green, blue and purple. Pour the remaining uncolored cheesecake mixture to the prepare crust and smooth out. Dollop the colored cheesecake mixtures all over the cheesecake in random patterns. Using a toothpick, swirl to create patterns. Be careful not to mix up too much or the colors could be muddy looking.
Once you are happy with your rainbow, place back in the oven for 30 to 35 minutes, until just barely golden at the edges and the middle has a very slight jiggle. Allow to cool completely at room temperature then keep covered and refrigerated.