Place a rack in middle of oven and preheat to 325°F. Pulse gingersnaps in a food processor until very fine crumbs form (you should have about 2 cups). Add flour, raw sugar, and salt and pulse to combine. Add butter and pulse until mixture is the consistency of wet sand.
Transfer mixture to a 9" pie dish. Using a measuring cup, press firmly onto bottom and up sides of pie dish. Place pie dish on a rimmed baking sheet and bake crust, rotating halfway through, until dry and set, 20–25 minutes. Transfer pie dish to a wire rack and let crust cool.
DO AHEAD: Crust can be baked 1 day ahead. Store wrapped tightly at room temperature.
Place gelatin and 2 tablespoons milk in a small bowl and let stand until gelatin is softened, 5–10 minutes.
Warm remaining 1/2 cup milk in a small saucepan over medium heat until steaming, about 3 minutes. Remove from heat, add gelatin mixture, and whisk until smooth. Whisk in yogurt, brown sugar, and lemon zest and pour into pie crust, smoothing top if needed; chill until set, at least 2 hours.
Meanwhile, bring grapes, granulated sugar, lemon juice, pepper, and salt to a boil in a large skillet over medium-high heat, stirring to dissolve sugar. Reduce heat and simmer, stirring occasionally, until grapes burst and mixture is syrupy, 8–10 minutes. Let cool.
Using a slotted spoon, spoon compote over pie slices just before serving.
DO AHEAD: Pie and compote can be made 2 days ahead; cover separately and chill.
