Preheat the oven to 400F. Line the tart pan with the dough, then line the dough with foil. Fill with ceramic pie weights or beans or rice. Place the tart shell on a baking sheet and bake until it starts to dry out, about 15 minutes. Remove the foil and weights and continue to bake until golden brown, about 5–10 minutes more. Remove from the oven and set aside to cool.
In a mixing bowl, whisk together the mascarpone, cream, vanilla, brown sugar, and salt until soft peaks form. Be careful, especially if you are using an electric mixer, because the mixture will thicken very quickly.
Carefully spread the mascarpone cream evenly onto the bottom of the tart shell.
Trim the stems off the figs and slice in half or quarters, lengthwise. Place them evenly on top of the mascarpone cream, overlapping so they all fit. Drizzle the figs with honey.