For the crust: Preheat oven to 350ºF. In food processor, pulverize cookies to fine crumbs (you should have 1 1/2 cups). Add brown sugar and cinnamon, and pulse briefly to mix. Turn machine on and, with machine running, drizzle melted butter through a feeding tube just until crumbs are evenly coated (mixture will not form a ball but remain crumbly).
Turn mixture out into a 9-inch springform pan. Using your fingers, evenly distribute crumbs over the bottom and up the sides of pan, gently pressing into an even thickness. Wrap foil around the outside of the bottom of the pan to help prevent any leaks from cheesecake. Bake on a rimmed baking sheet until lightly browned, 8 to 10 minutes. Cool on a rack.
For the cheesecake: Lower oven temperature to 250ºF. Put sweet potato in a microwavable container, cover with plastic wrap and microwave until very soft, 4 to 5 minutes. Measure out 1 1/4 cups packed brown sugar.
In a clean food processor, combine the cream cheese, ricotta, flour, and pumpkin pie spice. Puree, scraping down sides as necessary, until smooth (mixture will be fairly thick). Scrape mixture into a large bowl. In the processor (no need to clean it this time), combine 1 1/4 cups sweet potatoes, brown sugar, granulated sugar, and eggs. Puree, scraping sides as necessary, until well blended. Add to bowl and whisk until completely smooth.
Pour into prepared crust and set on a rimmed baking sheet, and then set on the center oven rack. Pour boiling water into baking sheet to about 1/2-inch up sides of cake pan. Bake until almost set (center will be slightly loose when jiggled), about 90 minutes. Remove pan from water bath and cool completely in pan on a rack. When completely cool, cover with foil and refrigerate 6 hours or overnight.
Tip: For a quicker pumpkin cheesecake, replace the sweet potatoes with 1 1/4 cups canned pumpkin.