Heat a large skillet over medium heat, then add chopped walnuts. Cook until fragrant, about 1-2 minutes, stirring often so they don't burn. Transfer to a bowl and set aside. Add 2 Tbsp olive oil to the skillet, then add anchovies. Use a wooden spoon or spatula to carefully press into and break apart the filets. Be careful, oil may splatter. Cook until the anchovies are melted and broken down, then pour into a mixing bowl.
Make the vinaigrette. To the bowl with the anchovy oil, add chopped herbs, vinegar, capers, Dijon, whole grain mustard, and sugar. Whisk to combine, then set aside.
Cook the French green beans. Add remaining 2 Tbsp olive oil to the skillet, then heat over medium-high heat. When hot, add trimmed green beans and cook, stirring often, until blistered (not burnt!), about 5-7 minutes. Green beans should still have a slight crunch to them. Season with kosher salt.
Add the vinaigrette. Pour the anchovy vinaigrette into the skillet along with the sliced scallions, then toss to combine. Transfer cooked green beans to a large serving bowl. Top with toasted walnuts, then serve immediately or at room temperature.