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Preheat oven to 350 degrees Fahrenheit. Invert a muffin tin and brush the bottom and sides of the four corner cups with a little of the melted butter.
Lay one sheet of filo dough on a dry surface. Brush with half the remaining butter.
Lay second sheet of filo on top. Brush with remaining butter. Trim one short end to make an even square. Cut square into four squares.
Place each square of filo over one of the buttered muffin cups. Gently fold and crimp filo around edges of cups to form an inverted bowl.
Place inverted pan in oven and bake 6 to 8 minutes, until filo is golden brown. Cool filo cups on pan. (the cups can be baked up to two days ahead and stored in an airtight container.)
Peel and core apples. Cut into 1/8- to 1/4-inch slices and toss with lemon juice in a medium bowl.
Combine sugar, butter, cinnamon, nutmeg, and 1/3 cup water in a wide skillet; bring to a rapid simmer over medium-high heat. Add apples and lemon juice. Cook 5 to 7 minutes, stirring frequently until apples are tender, adding 1 to 2 tablespoons of water if liquid evaporates. With a slotted spoon, remove apples to a bowl to cool slightly. Continue cooking liquid 1 to 2 minutes until syrupy.
Gently lift filo cups from pan. Spoon warm apples into cups. Place a scoop of ice cream over apples and drizzle with syrup. Serve immediately.