Preheat your oven to 350°F (175°C). Lightly coat a large 18- x 13-inch baking sheet with cooking spray.
Prepare the Filling: In a large pot or Dutch oven, heat 1 Tbs olive oil over medium heat.
Add the onion, bell pepper, 2 tsp (10 ml) minced garlic, and 2 tsp (10 ml) thyme.
Sauté for 10 minutes until softened and fragrant.
Add the sliced mushrooms and cook for another 10 minutes, allowing them to release their liquid and wilt.
Stir in the baby spinach and nutmeg, and season generously with salt and pepper.
Cover the pot and cook for 5 to 7 minutes, stirring occasionally, until the spinach is completely wilted.
Transfer the entire mixture to a colander and press firmly to drain as much liquid as possible.
Set aside in a large bowl to cool completely.
Make the Cheesy Binder: In a food processor, combine the drained tofu, softened Neufchâtel cheese, and grated Cheddar cheese. Purée until the mixture is completely smooth and creamy. Scrape this into the bowl with the cooled mushroom and spinach mixture and stir until well combined.
Season again with salt and pepper to taste.
Prepare the Garlic Oil: In a small saucepan, gently heat ¼ cup olive oil with 1 tsp (5 ml) minced garlic and 1 tsp (5 ml) thyme over medium-low heat for 2 to 3 minutes, just until fragrant. Be careful not to burn the garlic.
Remove from heat and let it cool slightly.
Assemble the Bottom Crust: Unroll your thawed phyllo dough.
Place two sheets of phyllo on the prepared baking sheet, overlapping them slightly in the center and allowing the edges to hang over the sides of the pan.
Brush the entire surface with the prepared garlic oil.
Repeat this process of layering and brushing with two sheets at a time until you have a 6-layer bottom crust (12 phyllo sheets total).
Add the Filling: Spread the filling evenly over the bottom crust, leaving a 3-inch border of phyllo dough all around the edges.
Brush this exposed border with garlic oil.
Create the Top Crust: Gently fold the oiled sides of the phyllo dough over the filling. Now, begin layering the top.
Place two sheets of phyllo over the filling, overlapping them in the center to cover it completely.
Brush with garlic oil.
Continue layering two sheets at a time and brushing with oil until you have used the remaining phyllo sheets (creating about an 8-layer top crust).
Finish and Bake: Carefully tuck all the overhanging edges of the phyllo dough under the strudel to create a neat package.
Brush the entire top surface with any remaining garlic oil and sprinkle with poppy, sesame, or fennel seeds, if using.
Bake for 45 to 50 minutes, or until the strudel is a deep golden brown and crisp.
Let it stand for at least 15 minutes before slicing and serving.
Keep your phyllo dough from drying out by covering the stack with a piece of plastic wrap and then a damp, clean kitchen towel while you work.