For the crust, put the flour, salt, and sugar in the bowl of a food processor with a blade. Pulse for a few seconds to combine. Add the butter and pulse some more, until the butter is in pea-sized pieces. Start by adding ⅓ cup of ice water, and pulse just until the dough starts to come together. Then add a little more if you need it (do not exceed ½ cup). Try not to over-pulse. Don’t worry if it doesn’t turn into a ball. Dump the crumbly mixture onto a sheet of plastic wrap, and use your hands and the wrap to help you slightly knead it and bring it together. Pat it into a rectangular flat disc. Wrap in the plastic and refrigerate for at least 1 hour.

Preheat the oven to 400 degrees F. Line a sheet pan with parchment paper. On a floured surface, roll the dough into a rectangular shape, as best as you can, until it is about ¼” thick. Don't make it too thin. Using a small knife, trim the edges (they might be over-squished by the rolling pin). Place the dough on the prepared sheet pan and refrigerate while you prepare the apples.

Peel the apples and cut around the cores. Save your peels and cores. Slice the apples in 1/4-inch thick slices (thin, but not too thin). Place overlapping slices of apples on the crust any way you want until it is completely covered with apple slices, even all the way to the edges (the apples will shrink in the oven). Sprinkle apples with the 1/3 cup of sugar and dot with the 4 tablespoons of butter. Depending on the thickness of your crust and hotness of your oven, bake the tart for 30-45 minutes, or just until the crust is cooked and the edges of the apples start to brown. Rotate the pan halfway during cooking. Start checking it after 30 minutes to see if it's done. The apple juices will burn a little, but don’t worry because that will happen.

Meanwhile, make the glaze: Put apple peels, cores, and any extra pieces (if you have some) in a saucepan, along with the sugar. Pour in just enough water to cover and simmer, uncovered, until it is reduced and thickened. Strain syrup through a sieve. When the tart's done, lightly brush with some of the glaze (you will have extra leftover). If you put too much glaze on it, it might be too sweet, so use a light hand. You can always add more. Allow to cool and serve warm or at room temperature.