When fully ripe, persimmons have a glossy red-orange skin, similar to tomatoes, and a tangy-sweet flavor. They are available at most markets from October through March; choose fruit that is plump and slightly soft but not at all mushy.
Heat broiler, with rack 6 inches from heat. Slice persimmons in half crosswise. Place halves cut side up in a baking dish, and drizzle each half with honey. Broil until tops are golden brown and caramelized, 6 to 8 minutes. Remove from oven; immediately drizzle 1 tablespoon lime juice over the tops.
While persimmons are broiling, whisk together mascarpone, vanilla, and remaining tablespoon lime juice in a medium bowl. Serve persimmons hot, each half topped with a dollop of mascarpone mixture and sprinkled with ginger or sugar, if desired.