Preheat oven to 350ºF. Line a baking tray with parchment paper. In a food processor, pulse the nuts, cinnamon and nutmeg. Set aside.

Place two sheets of thawed phyllo dough on a cutting board or clean countertop. Cover remaining phyllo with a clean, damp towel. Lightly brush the top sheet of phyllo dough with melted vegan butter.

Starting at one side, sprinkle about 1/4 cup of nut filling on the filo dough. Fold in about an inch on each side to prevent the nuts from falling out when rolling. Roll away from yourself, keeping a fairly firm grip so the filling is tightly packed.

Place the roll, seam-side down, on the prepared baking tray. Repeat until all the filo is gone and you have several rolls on the baking tray. With a sharp knife, cut the rolls diagonally into 2-inch pieces. Bake for 25 to 30 minutes. Baklava should be golden brown when done.

While the baklava is baking, prepare sugar syrup: Place a small pot over medium heat and mix together a the sugar and water. Simmer until sugar is dissolved, about 5 minutes. Keeping heat at a low simmer, add the lemon juice and vanilla. Let cool before using.

While the baklava is still hot, spoon syrup over all the pieces and and let the dough absorb it. Sprinkle a bit of leftover nuts on each piece for decoration and let rest for 45 to 60 minutes prior to serving. Best if you can let it sit overnight, but that may not be possible! Baklava can be stored at room temperature (in an air-tight container) for 2 to 3 weeks.