Bring a small pot of salted water to boil, then blanch beans for 2 to 3 minutes, until just tender. Transfer with tongs to a baking sheet to cool.
Make balsamic vinaigrette: Using a mortar and pestle, pound oregano, garlic, and a scant 1/4 tsp. salt to a paste (or chop garlic and oregano, sprinkle with the salt, mince, and mash into a paste with the flat side of a chef's knife). Transfer to a small bowl and pour in vinegars. Whisk in olive oil and taste for balance and seasoning.
Whisk hazelnut oil, lemon zest, and a couple of pinches of salt and pepper in a small bowl. Finely chop half of hazelnuts and stir into dressing; coarsely chop the rest and stir in. Drizzle hazelnut dressing over haricots verts, season with salt and pepper, and toss together.
Hold heirloom tomatoes on their sides and cut into 1/4-in.-thick slices. Season with salt and pepper. Arrange slices on a large round platter, overlapping them, and spoon on about half the balsamic vinaigrette. Scatter with arugula leaves.
Stir cherry tomatoes into remaining vinaigrette in bowl and season with salt and pepper. Pile in center of platter, then top with haricots verts. Spoon on a few dollops of crème fraîche and sprinkle about a third of pistou onto and around salad (serve the rest alongside).
*Find with the oils in well-stocked grocery stores.
Lucques, Tavern, Larder at Maple Drive, A.O.C., all in Los Angeles