Preheat your oven to 375°F (190°C). Generously butter a 9-inch pie dish or a similar-sized baking dish. Sprinkle the buttered dish evenly with 1 tablespoon of sugar, tilting to coat the bottom and sides.

Arrange the peach slices decoratively in a single layer across the bottom of the prepared dish. They can slightly overlap, but try to keep them relatively flat.

In a medium bowl, whisk together the 3 eggs and 1/2 cup of sugar until pale and frothy (about 2 minutes). Whisk in the vanilla extract.

Gently whisk in the flour until just combined, being careful not to overmix. Then, slowly whisk in the whole milk and the 1 tablespoon of melted butter until the batter is smooth. It will be thin.

Pour the batter evenly over the arranged peaches in the baking dish. The peaches will mostly sink, which is normal.

Bake for 40 to 45 minutes, or until the clafoutis is puffed up around the edges, golden brown, and a skewer inserted near the center comes out mostly clean (it should still look slightly custardy). Do not open the oven door during the first 30 minutes.

Let the clafoutis cool on a wire rack for at least 20 minutes before serving. Dust lightly with powdered sugar, if desired, and serve warm or at room temperature.