In a medium bowl, combine the finely chopped walnuts, cinnamon, nutmeg, and salt. Stir everything together until the walnuts are evenly coated with the spices. Set this mixture aside for later layering.
Preheat your oven to 325°F (165°C). Lightly brush the bottom and sides of a 9×13 inch baking pan with some of the melted butter. Unroll your thawed phyllo dough and keep it covered with a damp kitchen towel to prevent it from drying out as you work.
Place two sheets of phyllo into the bottom of the pan and brush the top sheet generously with melted butter. Repeat this until you have about 8 to 10 sheets layered at the bottom. Sprinkle about 3/4 cup of the walnut mixture evenly over the pastry.
Add 4 more sheets of phyllo, brushing every other sheet with butter. Sprinkle another layer of nuts. Continue this pattern of 4 sheets and a nut layer until you run out of nuts. End with a final top layer of 8 to 10 sheets of phyllo, brushing each one with butter.
Using a very sharp knife, cut the baklava into diamond or square shapes. Make sure to cut all the way through to the bottom of the pan. This allows the syrup to soak in later and makes serving much easier.
Place the pan in the oven and bake for 45 to 50 minutes, or until the pastry is a deep golden brown and looks crisp. While the baklava is baking, you can prepare the maple syrup glaze.
In a small saucepan, combine the maple syrup, sugar, and water. Bring to a boil, then reduce heat and simmer for about 10 minutes until slightly thickened. Remove from heat and stir in the vanilla extract. Let the syrup cool completely.
Immediately after removing the baklava from the oven, pour the cooled maple syrup evenly over the hot pastry. You should hear it sizzle! Let the baklava sit uncovered at room temperature for at least 4 to 6 hours (or overnight) to allow the syrup to absorb.