In a large bowl, toss together the butternut squash, carrots, parsnips, turnips, olive oil, smoked paprika and salt. Spread out in a single layer on a parchment-lined baking sheet. Roast, turning once or twice while cooking, until tender, 30-40 minutes.
Cook the lentils:
In a large pot, combine the lentils, water, and a pinch of salt. Set over high heat and bring to a boil. Lower heat to medium low and simmer, covered, until lentils are tender, 30-40 minutes. Drain and set aside.
Make the pesto:
Add the pecans, Parmesan cheese and olive oil to the bowl of a food processor. Pulse until crumbly. Season to taste with salt and black pepper. Set aside.
Assemble the strudel:
Preheat oven to 400°F.
Once all of the filling components are ready, remove the phyllo dough from its packaging and cover it with a damp cloth or paper towel. The phyllo dough dries out quickly, so it is important to keep it damp and work quickly. Place one sheet of phyllo on a parchment-lined baking sheet. Brush with the olive oil. Put another sheet of phyllo on top of it and brush with more oil. Repeat with remaining sheets.
Spread the roasted vegetables over the phyllo, leaving a 1-inch border around the edges. Sprinkle the lentils, pesto and crumbled goat cheese over the top. Fold the short sides of the dough in towards the center until they meet the filling. Repeat with the long sides and then roll the entire strudel lengthwise into a long log shape. Bake until golden brown, about 20 minutes. Let cool for 5 minutes before cutting into slices and serving.
The roasted root vegetables, lentils, and pecan pesto can all be made ahead of time. Just store them in the fridge until you are ready to stuff and roll your strudel!Phyllo dough can be a bit finicky, but the trick to make it easy to work with is to make sure it stays moist. Drape a damp cloth over the sheets whenever you are not actively using them.