Preheat oven to 350°. Toast hazelnuts on a rimmed baking sheet until browned and fragrant, 5 to 8 minutes.
Put squash in a microwave-safe container with 1/4 cup water and cover with plastic wrap. Cook on high until squash is tender when pierced, about 3 minutes. Drain.
Pulse bread in a food processor until fine crumbs form. Add squash and chickpeas and pulse until slightly chunky. Add egg, flour, sage, salt, pepper, and chile flakes. Pulse just until blended.
Heat 3 tbsp. oil in a large frying pan over high heat. Drop 1/4-cup portions of squash mixture into oil, making a few fritters at a time, and cook, turning once, until golden brown, about 3 minutes on each side. Transfer to a platter.
Toss greens with remaining 3 tbsp. oil, the lemon juice, and hazelnuts. Serve squash fritters with salad.