Combine the cookies and sugar in a food processor and pulse to make fine (but not pasty) crumbs. Drizzle in the melted butter and pulse to combine. Press the crust into the bottom and up the sides of a 9-inch tart pan with removable bottom. Bake on a baking sheet until the crust is set and just golden, about 12 minutes. Cool completely.
For the filling: Combine the chocolate, cream, butter and dried ginger in a small saucepan and cook over low heat, stirring until the chocolate melts, about 5 minutes. Stir until very smooth. Pour into the cooled tart shell and refrigerate until set, at least 3 hours.
For the salted caramel: Combine the sugar, corn syrup and 1/4 cup water in a small saucepan. Set the pan over medium-high heat and cook, stirring occasionally, until the sugar is a deep amber, 8 to 12 minutes. Carefully whisk in the cream until smooth. Remove from the heat and stir in the butter, vanilla, kosher salt and cayenne. Pour into a spouted glass measuring cup and let cool until thickened, 1 to 1 1/2 hours.
When the caramel is cooled and the tart is set, drizzle the caramel over the tart so some of the chocolate is still exposed; save the rest for serving. Sprinkle the tart with flaky sea salt and chopped crystallized ginger. Cut into wedges. (If you make the tart ahead of time, use a warm knife or let sit at room temperature for a few minutes before slicing.)