Blanch and peel peaches; cut them in half; remove the pits, and slice them coarse.
Note: to blanch peaches—have ready a large bowl of ice-cold water, a slotted spoon, and a saucepan of boiling water deep enough to cover the peaches; with the slotted spoon place the peaches in the boiling water, two or three at a time (if the peaches are fully ripe, they will need only about 15 seconds in the boiling water, if not quite ripe, they will need more); with the slotted spoon, raise a peach from the water and move your thumb firmly over the skin; if the skin has loosened enough it will wrinkle and feel loose; at that point transfer it to the ice water; peel with your fingers, starting at the stem end.
Place sliced peaches in a 3-quart saucepan with the sugar, butter, and water; stir briefly over med-high heat just until the mixture comes to a boil; remove from the heat and set aside.
Adjust oven rack to the middle of the oven; preheat to 375°.
Generously butter a shallow ovenproof baking dish with at least a 2-quart capacity; it can be larger but not smaller.
Sift together the flour, baking powder, and salt; set aside.
In a small bowl, beat the butter with the vanilla and almond extract; beat in the sugar to mix.
On low speed add in half of the milk, then half of the sifted dry ingredients, the remainder of the milk then the remainder of the dry ingredients, scraping the sides of the bowl with a rubber spatula as needed and beating only until mixed.
Pour the batter into the baking dish and spread it smooth.
With a large slotted spoon, place the prepared peaches evenly over the batter (reserve the peach syrup for the moment); place the dish in the oven.
Now, pouring the peach syrup (it will be thin) over the fruit will be easier if you pour it in two additions into a 2-cup measuring cup, then reach into the oven and quickly pour it over the top of the fruit in the baking dish.
Place a cookie sheet on a rack below the baking dish to catch spills.
Bake for 20 minutes; then decrease temperature to 325° and bake for 25 minutes.
Serve immediately or within 20-30 minutes on large flat plates or in dessert bowls, spooning some of the thickened syrup from the bottom over or alongside each portion.