Start by finely chopping the pecans and walnuts. In a bowl, combine the chopped nuts with the ground cinnamon and nutmeg. Stir well so the spices coat the nuts evenly.
In a small saucepan, combine the maple syrup, honey, water, and vanilla extract. Bring to a gentle simmer over medium heat, stirring occasionally. Let it cook for about 10 minutes until it slightly thickens. Remove from heat and set aside to cool.
Preheat your oven to 350°F (175°C). Lightly grease your baking dish with some melted butter. Carefully unroll the thawed phyllo dough and cover it with a damp clean towel to prevent drying out. Place one sheet of phyllo in the baking dish and brush it with melted butter. Repeat this process, layering and brushing each sheet with butter, until you have about 8 sheets layered.
Evenly spread half of the nut mixture over the layered phyllo. Then, layer 4 more sheets of phyllo on top, brushing each with butter as before. Add the remaining nuts and finish by topping with the remaining phyllo sheets, continuing the butter-brushing process.
With a sharp knife, carefully cut the assembled baklava into diamond or square shapes. Bake in the preheated oven for 45 to 50 minutes, or until the phyllo is golden and crisp.
Once the baklava is out of the oven, immediately pour the cooled maple-honey syrup evenly over the hot baklava. Let it soak at room temperature for several hours or overnight to fully absorb the syrup.