Ahead of time (the biggest color trick): Spread the apple quarters cut-side up on a rack, uncovered, in the fridge for a few hours or overnight. If you are short on time, pat them very dry. Make the pastry, roll it to ⅛ inch, cut a 10-inch round, and chill it on parchment.
Set up: Put an oven rack in the lower-middle position and heat to 400°F with a baking sheet on the rack below. Brush the pan’s sides with the soft butter and keep it off the bottom.
Make the caramel: Put the sugar and water in the pan over medium heat. Stir once to wet the sugar, then stop stirring and swirl. In 6 to 9 minutes it should turn a deep amber, about 340°F, like an old penny. Pull it a shade lighter than you want, since cast iron holds heat. Off the heat, carefully swirl in the butter pieces, because it spatters. If it separates, whisk over low heat for 30 seconds.
Pack the apples: Stand the slices on edge, rounded side down, in tight concentric circles with the extra quarter in the center. Pack them so they cannot tip. They shrink a lot.
Cook on the stove: Cook over medium heat (not medium-low), uncovered, 20 to 30 minutes. Baste with a spoon every 5 minutes and give the pan a quarter turn so hot spots even out. The syrup will bubble thin and pale at first, then thicken and darken. Move on when it is thick and deep brown and a spoon lifted at the edge shows mahogany underneath. If it smells bitter, take it off the heat.
Add the pastry: Let the pan cool just 3 to 5 minutes so the caramel stays fluid. Lay the cold pastry over the top, tuck the edges down inside the rim, and cut a few vents.
Bake: Set the pan on the baking sheet and bake 25 to 30 minutes for puff pastry or 30 to 35 for brisée, until deep golden.
Flip while hot: Rest it only 3 to 5 minutes out of the oven. Run a thin knife all the way around the edge. Set the rimmed plate on top, invert in one quick motion with mitts, and wait 30 seconds before lifting the pan slowly so the caramel flows down.
If it sticks: Put the pan back on a low burner for 30 to 45 seconds to re-melt the caramel, then retry. If apples stay behind, nudge them back into place and spoon the pan caramel over the top.