Make the Simple Syrup (Agda). In a medium pot, mix water, sugar, and vanilla. Turn the heat to high and bring it to a boil. Once boiling, lower the heat to medium. Add lemon slices and cook for 5–7 more minutes. Remove from heat and let it cool.Do the next steps while the syrup is cooking. Make the Walnut Filling. Blend half the walnuts until flour consistency. Put them in a large bowl. Chop the rest of the walnuts into small pieces (use a food processor, blender, or knife), then add them to the bowl. Add sugar and mix everything well.
Melt the Butter. In a small pot, melt the butter on low heat. When fully melted, add oil and whisk until smooth.
Cut the Phyllo to Fit the Pan. Unroll half of the phyllo sheets on your work surface. Place your baking pan on top of them. Cut around the pan so the phyllo fits its shape (use a knife or pizza cutter). Discard the extra pieces. Repeat with the other half of the phyllo sheets. Set aside 3–4 sheets for the final top layer. Cover those with plastic wrap or a towel so they don’t dry out. Preheat the oven to 300°F (150°C). Make the Bottom Layer. Brush the baking pan with butter. Add one sheet of phyllo to the bottom, then brush it with butter. Repeat this until you’ve layered six phyllo sheets, brushing butter between each one.
Layer the Baklava. Take a large handful of filling (about 1/11 of it) and spread it evenly over the phyllo using your hand. Add two phyllo sheets on top, brushing each with butter. Repeat this process—filling, then two buttered phyllo sheets—until all the filling is used. Finish with the 3–4 top sheets you set aside, brushing each with butter.You’ll end up with about 10 layers of filling in between the bottom and top layers. Cut the Baklava. Use a sharp knife to cut the baklava into long vertical strips, about 1.5 inches (4 cm) wide. Then rotate the pan 45 degrees and cut vertical strips again from this side to form diamond shapes. A ruler helps keep the cuts even.
Bake the Baklava. Place the pan on the middle oven rack. Bake for 15 minutes, or until the top turns golden. Lower the heat to 265°F (130°C) and continue baking for 90–120 minutes. Check on it occasionally. It’s done when the baklava slightly 'moves' (shifts) in the pan when you gently shake it side to side. It should be a nice golden brown. You can also gently lift one piece with a fork to check if the bottom is baked.
Soak with Syrup. Take the pan out of the oven and let it cool for 10 minutes. Use a ladle to slowly pour the cooled syrup over the baklava. Start from the edges and move toward the center. Pour half first, wait a minute for it to soak in, then pour the rest. Place lemon slices evenly on top. If you have a flat, slightly heavy object larger than the pan (like a serving tray), place it on top for 10–15 minutes. Then remove it and cover the pan with a kitchen towel.
Let It Rest. Leave the baklava in a cool room (but not the fridge) for several hours, or overnight, so it can fully absorb the syrup.