Step 1Place a rack in middle of oven; preheat to 400°. Arrange 6–8 thin slices prosciutto on a parchment-lined baking sheet, overlapping by no more than ½” if needed, and bake until crisp, 7–9 minutes. Let cool.
Step 2Meanwhile, cook 2 lb. green beans, trimmed, in a medium pot of boiling generously salted water until bright green and crisp-tender, about 4 minutes. Drain beans; transfer to a large bowl of ice water and let cool. Drain green beans and thoroughly pat dry with a kitchen towel or paper towels. Set aside.
Step 3Cook ⅔ cup extra-virgin olive oil and ½ cup coarsely chopped blanched hazelnuts in a medium skillet over medium heat, stirring often, until hazelnuts are golden brown, 5–7 minutes. Immediately transfer nuts and oil to a large bowl; reserve skillet. Let cool until just barely warm, about 10 minutes.
Step 4While nuts and oil are cooling, cook 12 Medjool dates, pitted, quartered lengthwise, in reserved skillet over medium-high heat, tossing occasionally and adding up to 2 tsp. more extra-virgin olive oil if pan is dry, until lightly charred in spots, about 2 minutes. Transfer dates to a plate.
Step 5Add 1 large shallot, finely chopped, 3 garlic cloves, finely chopped, zest of 1 medium lemon, ¼ cup fresh lemon juice, ¼ cup red wine vinegar, 2 Tbsp. honey, 2 tsp. Dijon mustard, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt to bowl with nuts and oil and whisk to combine; generously season vinaigrette with freshly ground pepper.
Step 6Add dates and reserved green beans to vinaigrette and toss to coat. Taste and season with more salt and pepper if needed. Transfer green bean mixture to a platter or large shallow bowl. Crumble prosciutto into small shards and scatter over green bean mixture.
Do Ahead: Prosciutto can be baked 4 hours ahead; store, loosely covered, at room temperature. Green beans can be blanched 1 day ahead; pat dry, transfer to an airtight container and chill. Nuts can be cooked 1 day ahead; transfer to an airtight container along with oil and store at room temperature.