Preheat your oven to 375°F (190°C). Generously butter a 9-inch pie dish or a shallow gratin dish with the 2 tablespoons of melted butter, ensuring the sides are coated.

Arrange the halved and pitted apricots in a single, slightly overlapping layer across the bottom of the prepared baking dish.

In a medium bowl, whisk together the flour, sugar, and salt. Gradually whisk in the milk until the mixture is smooth. Whisk in the eggs one at a time until fully incorporated, followed by the vanilla extract.

Carefully pour the liquid batter evenly over the apricots in the baking dish. The fruit may float slightly; this is acceptable.

Bake for 35 to 40 minutes, or until the flaugnarde is puffed, golden brown around the edges, and a knife inserted near the center comes out mostly clean (a slight custardy tremble is okay).

Remove from the oven and let cool on a wire rack for at least 15 minutes before serving. Dust generously with additional granulated sugar just before serving warm.