4 pounds honeynut or butternut squash, halved lengthwise (from 2 to 3 honeynuts or 1 large butternut)
2 tablespoons extra-virgin olive oil, plus more for frying and drizzling
Kosher salt and freshly ground pepper
1 pound parsnips (4 to 5 medium), peeled and halved lengthwise
2 pounds leeks (3 medium), white and light-green parts only, halved lengthwise and thoroughly washed and drained
1 tablespoon fresh thyme leaves
1 tablespoon chopped fresh sage leaves, plus whole leaves for frying
2 Granny Smith apples (1 pound), halved and cored
1 quart low-sodium chicken broth
1 stick unsalted butter
Toasted pepitas, toasted sesame seeds, and poppy seeds, for serving
Preheat oven to 400 degrees, with racks in upper and lower thirds. Scoop seeds and pulp from squash; discard. (Or lightly coat seeds in oil, season with salt, and roast on a rimmed baking sheet until crisp and darkened slightly, about 20 minutes; let cool and reserve for garnish.) On a rimmed baking sheet, rub squash halves with 1 tablespoon oil and season with salt; turn cut-sides down. On another rimmed baking sheet, toss parsnips and leeks with remaining 1 tablespoon oil, season with salt, and sprinkle evenly with thyme and chopped sage; spread in a single layer.
Roast 30 minutes. Add apples to sheet with squash, cut-sides up. Continue roasting until vegetables turn golden brown in places and are easily pierced with the tip of a knife, 15 to 20 minutes more. When cool enough to handle, scoop flesh from squash; transfer half to a blender with half of other vegetables and apples, 2 cups broth, and 1 cup water. Puree until smooth, adding more water as needed if too thick to self-level. Pour through a sieve into a pot. Repeat process with remaining vegetables, apples, broth, and 1 more cup water.
Melt butter in a small saucepan over medium heat, swirling occasionally, until fragrant and golden brown and dark-brown sediment particles form in bottom of pan, 8 to 10 minutes. Stir brown butter into soup; season with salt and pepper. Rewarm soup over medium-low heat, stirring occasionally and adding more water as necessary until it reaches desired consistency.
Wipe pan clean. Heat 1/4 inch of oil over medium-high. When it shimmers, add a handful of sage leaves; cook, stirring a few times, until darkened slightly, 20 to 30 seconds. Transfer to paper towels, season with salt, and let stand until cool and crisp, 2 to 3 minutes. Fry more sage as desired. Serve soup topped with crisped sage, pepitas, sesame seeds, poppy seeds, and a drizzle of oil.