First, make the syrup. In a medium saucepan, combine sugar, lemon slices, and water; bring to a boil. Reduce heat and simmer until sugar dissolves and the mixture is syrupy about 10-15 minutes. Set aside to cool.
Next, in a food processor pulse walnuts until finely chopped or ground. You can also use a rotary grater if you have one. In a large bowl, combine walnuts with cinnamon, cardamom if using, and sugar.
Unroll fillo dough. Trim the whole stack, if necessary, to fit the 13x9-inch pan. The easiest way to do this is with a pair of scissors. Cover fillo sheets with a clean cloth to keep from drying out as you work. If you work fast enough, you don't need to cover it.
Preheat oven to 350F. Butter the bottom and sides of a pan.
Place 1 sheet of dough in prepared baking pan; using a pastry brush, butter thoroughly. Repeat with 4 more sheets of fillo, laying each on top of the other. Sprinkle with ⅓ cup of walnut filling. Add two sheets of fillo dough, buttering each sheet, then sprinkle with ⅓ cup of nuts. Repeat until filling is used up, finishing with 5 sheets of fillo dough on top. So it goes like this: 5 buttered sheets + filling, 2 buttered sheets + filling, repeat..., finish with 5 buttered sheets.
Using a sharp knife cut the unbaked baklava into squares or diamonds all the way to the bottom of the pan. Pour remaining butter over the whole baklava and lightly sprinkle the top of pastry with cold water. This inhibits the pastry from curling.
Finally, bake baklava for 50-60 minutes or until golden and crisp. Remove from the oven and immediately spoon syrup over it.
Let cool for at least 4 hours.
Variations: Substitute half of the walnuts with pistachios.
Use butter ghee instead of butter.
Add 2 tablespoons honey to the syrup.
Add 1 tablespoon rose water to the cooked syrup.
Always pour cold syrup over hot baklava.
If using Fillo Factory fillo, layer 3 sheets on the bottom and top as they are a bit thicker than other brands'.