Grease a 10-inch springform pan with 1 teaspoon of the butter.
Combine the crumbs, 2 tablespoons of the sugar, and the melted butter in the bowl of a food processor and mix on low speed until well blended. Transfer to a greased 10-inch springform pan and press evenly around the bottom. Chill in the refrigerator for 1 hour.
Preheat the oven to 325 degrees F.
Combine the remaining 1/2 cup sugar and the water in a heavy, medium saucepan, and bring to a boil. Reduce the heat to medium and cook, stirring occasionally, until the sugar is dissolved. Continue cooking, swirling the pan occasionally, until the caramel is amber in color, 5 to 7 minutes. Add the remaining 4 tablespoons butter, stir, and cook over low heat until the butter is melted and evenly blended. Add the apples and stir to coat with the caramel . Cook, stirring, over medium heat until the apples are tender, about 5 minutes. Remove from the heat and let cool.
Combine the cream cheese and brown sugar in a large bowl and beat with an electric mixer until smooth. Add the eggs 1 at a time, beating well after each addition and scraping the sides of the bowl as necessary. Add the lemon juice , cinnamon, and nutmeg and mix well. Fold the apple-caramel mixture into the batter , and pour into the chilled pan. Bake, rotating the pan a quarter turn every 30 minutes, until golden brown and firm around the edges, but still slightly loose in the center, 1 hour and 20 minutes to 1 hour and 30 minutes.
Remove from the oven and cool for 2 hours. Cover the cheesecake and refrigerate for at least 4 hours, or overnight.
To serve, spoon a portion of the Spiced Creme Anglaise onto each serving plate, top with a slice of the cheesecake, and drizzle with the Butterscotch Sauce.
Prepare an ice bath by filling a large metal bowl with ice and water.
Combine cream, zest, cinnamon , and nutmeg in a medium saucepan and bring to a simmer over medium heat. Remove from the heat.
In a medium bowl, beat the egg yolks and sugar until pale and thick, about 2 minutes. Slowly drizzle in the hot cream, whisking constantly. Return the mixture to the pan and stirring constantly, cook over medium heat until the sauce thickens and coats the back of a spoon.
Place in an ice bath and stir until cool.
Cover with plastic wrap , pressing down against the surface to prevent a skin from forming. Refrigerate until well chilled, about 2 hours. Serve cold.
Yield: 2 cups Combine the sugar, water, and lemon juice in a medium, heavy saucepan and cook over medium-high heat, stirring, until the sugar dissolves. Let boil without stirring until the mixture becomes a deep amber color, 3 to 4 minutes, watching closely so it doesn't burn. Remove the pan from the heat. Add the butter and swirl the pan to melt it, about 1 minute. Return to the heat and bring to a boil. Add the cream, whisk to combine, and remove from the heat. Add 1/4 cup of the milk , then add up to 1/4 cup more until the desired consistency is reached.
Cool until just warm before serving. (The sauce will thicken as it cools.)
Yield: 2 cups
