Bringing the cream cheese, eggs, and sour cream to room temperature before blending results in a silky-smooth batter without any lumps.
Preheat oven to 350 degrees. Finely grind crackers in a food processor. Add melted butter, brown sugar, zest, and 1/2 teaspoon salt; pulse until texture is like wet sand. Press evenly into bottom and halfway up sides of a buttered 9-inch springform pan. Bake until set, 12 to 15 minutes. Let cool.
Reduce temperature to 325 degrees. Beat cream cheese on medium speed until smooth. Gradually beat in granulated sugar until light and fluffy, 2 to 3 minutes. Beat in lemon juice and remaining 3/4 teaspoon salt. Beat in eggs, one at a time, then sour cream and vanilla until smooth.
Place pan in center of a double layer of foil. Lift edges of foil up, wrapping it tightly around sides of pan and folding it in under itself as necessary until flush with top of pan. (This prevents water from seeping in.)
Pour filling into pan (it should come right to top of rim) and smooth top with a small offset spatula.
Place springform pan in a roasting pan; transfer to oven. Pour enough boiling water into roasting pan to come halfway up sides of springform pan. Bake until cake is puffed and golden brown on top and slightly wobbly in center, 1 hour, 45 minutes to 2 hours.
Remove springform pan from roasting pan; let cool on a wire rack 20 minutes. Remove foil and run a paring knife around sides of pan to loosen. Let cool completely. Drape pan with plastic wrap; refrigerate until cold, at least 8 hours and up to 3 days. Remove cake from pan; serve.