1 tbsp extra-virgin olive oil
PREHEAT the oven to 400°F. Line a rimmed baking sheet with parchment paper. Cut the spaghetti squash in half lengthwise through the middle. Scoop out the seeds with a large spoon and brush the flesh and skin with the olive oil. Sprinkle the flesh lightly with salt and place the halves cut side down on the prepared baking sheet. Roast until the squash is tender but al dente when pierced with a fork, 30 to 40 minutes. Let the squash cool to the touch. Use a fork to pull the squash strands from the skin, using the side of the fork to scoop the flesh closest to the skin; transfer to a large bowl and set aside.
MELT the butter in a large skillet over medium heat, then cook, swirling the pan frequently, until lightly browned, 2 to 3 minutes. Carefully pour out the butter into a bowl, allowing any burnt solids to stay in the bottom of the skillet. Wipe the skillet clean with a couple of folded paper towels, then return the brown butter to the skillet and heat it again over medium heat. Add 1 piece of sage to test the butter for readiness: It should immediately sizzle. When the butter is hot enough, scatter the remaining sage into it and cook, turning the leaves often with a slotted spoon, until they are crispy, 20 to 40 seconds. Using a slotted spoon, transfer the fried sage to paper towels to drain. Season it lightly with salt.
ADD the spaghetti squash to the brown butter, and adjust the heat to high. Add the lemon juice, sprinkle the squash with ¼ tsp of salt and the 1/8 tsp of pepper, and cook, turning the strands in the butter with tongs, until lightly browned on some edges, 4 to 6 minutes. Add the hazelnuts and two thirds of the sage and remove from the heat. Toss the spaghetti squash and transfer it to individual bowls or a large serving bowl, twisting the tongs to plate the squash strands into a tight nest. Top generously with freshly grated parmesan, grinds of pepper, and the remaining fried sage.