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1
Preheat the oven to 325 degrees F with a rack in the lower third of the oven.
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2
To make the crust, pulse the cookie crumbs, chopped pecans, and 1 tablespoon of the sugar in a food processor until evenly fine. Melt 1 1/2 tablespoons of the butter, add to the crumbs, and pulse until well mixed.
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3
Distribute the crumbs over the bottom of a 9-by-3-inch springform pan (ungreased). Press the crumbs into a solid layer covering the bottom, only. Bake the crust for 15 minutes -- it should be slightly darker than when you put it in. * Leave the oven on. *
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4
Let cool on a wire rack to room temperature. When the pan is still slightly warm, use a pastry brush to butter the sides of the pan.
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5
To make the caramelized apples, heat the remaining 4 tablespoons (1/2 stick) of butter with 6 tablespoons of sugar in a large skillet over medium heat, stirring and swirling, until the mixture is smooth and turns light amber. Add the apples, ginger, and a pinch of salt and continue to saute, turning to coat the apples, until they are tender but not mushy and are bathed in a golden caramel. Remove from the heat and distribute half of the apples evenly over the crust. Transfer the remaining apples and caramel to a bowl and refrigerate, covering with plastic wrap after it is cold.
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6
To make the filling, beat the cream cheese and the remaining 1/2 cup sugar in a standing mixer using the paddle attachment until smooth and no lumps remain. Add the lemon juice, vanilla, and a good pinch of salt, mixing until combined. Add the eggs, one at a time, scraping the bowl between additions and mixing just long enough to combine. (You can also mix the filling in a food processor. Most important is to get everything perfectly smooth without beating in air, especially after the eggs are added.)
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7
Carefully wrap 2 layers of heavy duty aluminum foil over the bottom of the pan, pressing it tightly onto the bottom and part way up the sides of the pan. (This prevents water from seeping in while baking.) Have on hand a larger baking pan that is about 2 inches deep. Fill a kettle with hot water.
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8
Pour the filling over the apples in the pan and tilt or spread to level it. Tap the pan on the counter a couple of times to release any air bubbles. Place the cake into the larger pan, place the whole thing on the oven rack, and pour hot water from the kettle into the outer pan to reach about half way up the sides of the springform pan.
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9
Bake until the cake moves in a solid jiggle when nudged, about 45 minutes. The center 2 inches or so will not seem completely cooked through. Remove the pan from the water bath and, working carefully with oven mitts over the sink, remove the foil to release any trapped water.
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10
Cool the cake in its pan on a wire rack until it is completely cool, about 3 hours. Run a knife or metal spatula around the sides of the pan, going all the way to the bottom. Unlatch and remove the sides of the pan, transfer the cake on its base to a plate, and refrigerate at least 8 or up to 24 hours.
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11
Remove the bowl of reserved apple topping from the fridge about an hour before serving the cake. Slide a flat flexible metal spatula between the crust and the pan bottom to loosen it, then carefully slide the cake from the pan bottom directly onto a serving plate. (If that makes you nervous, just transfer the cake on its pan bottom to a serving plate.) Distribute the apple topping evenly over the cake and decorate with toasted pecan halves if you like.
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12
To serve, dip the blade of a thin, sharp knife into hot water and shake off the water before cutting the cake in straight downward strokes, pulling the knife out toward the edge of the cake (not back up!). Wipe or rinse and shake off the knife as needed between cuts. Refrigerate any leftovers, tightly covered, for up to 1 day.
