Combine the prunes, rum, and hot black tea in a bowl. Soak until plump, 20–30 minutes; drain and pat dry.
Melt the butter. Brush a 9x13-inch baking dish with a little of the melted butter; reserve the rest. Preheat the oven to 375°F (190°C).
In a large bowl, whisk the flour, sugar, and salt to combine.
Whisk in the eggs one at a time until smooth. Gradually whisk in the milk, then the vanilla. Whisk in the remaining melted butter.
Let the batter rest at room temperature for 30 minutes to relax the flour and reduce bubbles.
Spread the drained prunes evenly in the buttered dish. Give the batter a brief whisk, then pour it over the prunes; tap the dish gently to release any large bubbles.
Bake until puffed, deeply golden, and just set in the center with a slight wobble, 45–55 minutes. If browning too quickly, tent loosely with foil.
Cool until warm or room temperature before slicing. Serve plain or chilled; the custard will firm as it cools.