In a non-reactive bowl, whisk together all of the ingredients for the hazelnut vinaigrette but the oils. Slowly drizzle the oils while whisking to emulsify. Reserve.
Prepare an ice bath. Remove the stems and strings from the yellow wax beans and haricots verts, leaving a 1-inch tail for aesthetic reasons. Bring 2½ quarts water and salt to a rapid boil. Blanch the beans. Immediately shock beans in the ice bath, retaining bright color, texture and flavor. When cooled through, remove beans from ice and keep chilled until service. Beans may be blanched in the morning and refrigerated, but they should be cooked as close to service as possible.
Preheat oven to 350 degrees. Roast the hazelnuts for 10 minutes or until they are fragrant and take on color. Remove from the oven and dump hazelnuts onto a cooled sheet pan. Rub off approximately half of the skins with a clean dish towel and reserve.
Meanwhile, heat a cast iron pan over medium-high heat. Add the onions and sear. Transfer the onions to a sheet pan and finish baking in the oven at 350 degrees until soft (about 8-10 min). Remove from the oven, allow to cool, then skin and store in olive oil or a portion of the vinaigrette for up to 1 week.
Heat grill to medium-high. Peel the thick outer layer from the fennel bulb and carefully slice on a mandoline, approximately 2mm thick. Reserve the slices in acidulated ice water (ice water with 1 Tbsp of lemon juice). When ready to grill, remove the fennel from the water and pat dry. Reserve the water. Lightly toss in a bowl with 2 tsp of the olive oil and black pepper. Place the fennel on the hottest part of the grill and allow the fennel to smoke for approximately 1 minute. Turn and cook for an additional 30 seconds. Remove the fennel and quickly plunge into the existing ice water to arrest cooking and refresh. Remove from the water, pat dry and toss in a bowl with the remaining lemon juice and olive oil. Season with salt and pepper to taste. Reserve.
To serve, toss all of the ingredients together in a non-reactive bowl and season this yellow wax beans salad with salt and pepper. Garnish with additional nasturtium petals, basil and fleur de sel.