Heat butter in a small saucepan over medium-low heat until it smells nutty and brown bits form on the bottom, about 15 minutes.

Pulse graham crackers, brown sugar, and salt in a food processor until fine crumbs form. Carefully pour browned butter into crumb mixture while food processor is running.

Press combined mixture into bottom and sides of a 10-inch tart pan with a removable bottom. Freeze for 20 minutes or chill for 1 hour until set.

Stir gelatin and cold water together in a small bowl. Let stand for 5 minutes, then microwave until gelatin dissolves, about 10 seconds.

Beat cream cheese in a bowl with an electric mixer for 30 seconds. Add yogurt, white sugar, lemon juice, lemon zest, salt, and vanilla. Beat until smooth, then beat in gelatin.

Pour mixture into the chilled crust. Chill until set, about 2 hours.