Preheat oven to 400° with a rack in the top third. Put hazelnuts on a rimmed baking sheet. Line another rimmed baking sheet with parchment paper.Roast hazelnuts until slightly darker and skins begin to blister, about 7 minutes. Wrap in a kitchen towel and let sit 2 to 3 minutes. Rub nuts against one another in towel until most of the skins are off (it's okay to have some left). Set nuts aside and let cool completely. Once cool, coarsely chop 3 or 4 nuts for topping.
Meanwhile, put bell peppers in a large bowl. Drizzle with 1/2 tbsp. oil, sprinkle with a generous pinch of salt, and toss to coat well. Transfer to lined baking sheet and roast 15 minutes. Turn peppers over and roast until charred, soft, and skins are beginning to puff and/or peel away, 20 minutes more. Transfer to a bowl, cover, and let sit until cool enough to handle. Remove and discard stems, skins, seeds, and juices. Chop peppers roughly and set aside.
Heat 1 tbsp. oil in a medium frying pan over medium heat. Add garlic, shallot, and chile flakes and cook, stirring, until garlic is just beginning to brown and shallot has softened, about 2 minutes.
In a food processor, pulse toasted whole hazelnuts until chopped. Add roasted peppers and garlic-shallot mixture to processor and pulse a few more times, scraping processor with a spatula if necessary. Add remaining 1 tbsp. oil, the sherry vinegar, 1/2 tsp. salt, and the pepper. Process to a coarse but uniform purée.
Stir in mint and lemon juice, then season with more salt and pepper if you like. Spoon into a serving bowl and sprinkle with some reserved chopped hazelnuts.
Make ahead: Up to 1 week, chilled airtight.