Grate chilled butter on the large holes of a box grater and toss with flour, sugar, and salt until coated; sprinkle ice water a tablespoon at a time and mix until dough holds when squeezed, forming a shaggy dough.

Form the dough into a 6-inch disk, wrap tightly, and chill at least 20 minutes (or up to 48 hours).

Slice peaches and combine in a saucepan with sugar, lemon juice, vanilla, and cornstarch; warm over medium-low until sugar dissolves and juices thicken slightly, 3–5 minutes, then remove from heat and cool briefly.

Preheat oven to 400°F and line a baking sheet with parchment. Roll the chilled dough to a 12-inch circle on a lightly floured surface and transfer to the parchment-lined sheet.

Sprinkle almond flour or polenta in the center leaving a 2-inch border; arrange cooled peach slices in overlapping circles on top of the almond flour.

Fold the 2-inch border up over the fruit in pleats, pressing gently to seal, and brush the dough edge with egg wash; sprinkle coarse sugar over the crust.

Bake at 400°F for 35–45 minutes until the crust is deep golden and filling bubbles gently; rotate the pan once if needed for even browning.

Let cool on the sheet for at least 20 minutes; brush exposed peaches with warmed apricot jam if desired before slicing and serving.