Heat a medium skillet over medium-high. Combine 16–20 sage leaves (from about 4 sprigs), coarsely torn, and ½ cup coarsely chopped blanched hazelnuts in pan and drizzle 2 tsp. extra-virgin olive oil over. Sprinkle with ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt and season with freshly ground black pepper. Toast, stirring often, until nuts are golden and sage is crisp, about 4 minutes. Transfer to a small bowl. Wipe out skillet and reserve.

Place 1 small garlic clove, finely grated, in a medium bowl. Heat 4 Tbsp. unsalted butter in reserved skillet over medium-high, swirling often, until melted and foaming. Add 1 medium shallot, finely chopped, and reduce heat to medium. Cook, stirring occasionally, until butter turns golden brown and smells nutty and shallot is lightly golden, about 3 minutes. Pour brown butter mixture into bowl with garlic, scraping up browned bits stuck to bottom of pan. Add 2 Tbsp. apple cider vinegar, 1 tsp. Dijon mustard, and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt and whisk to combine; season with pepper. Set vinaigrette aside.

Heat 1 Tbsp. extra-virgin olive oil in a large skillet, preferably cast-iron, over medium-high. Add 1 lb. green beans, trimmed, sprinkle with 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and season with pepper. Cook, tossing occasionally, until bright green and charred in spots but still crisp-tender, 5–7 minutes. Transfer to a large plate. Repeat process with remaining 1 lb. green beans, trimmed, 1 Tbsp. extra-virgin olive oil, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and more pepper. Remove from heat and return all of the green beans to skillet (if they don’t fit, use a large bowl instead). Add reserved vinaigrette and toss to coat.

Transfer dressed green beans to a platter and top with hazelnuts and sage.