Heat a large skillet over medium heat, then add chopped walnuts. Cook until fragrant. Transfer to a bowl and set aside. Add olive oil to the skillet, then add anchovies. Cook until the anchovies are melted and broken down, then pour into a mixing bowl.
To the bowl with the anchovy oil, add chopped herbs, vinegar, capers, Dijon, whole grain mustard, and sugar. Whisk to combine, then set aside.
Add remaining olive oil to the skillet, then heat over medium-high heat. When hot, add trimmed green beans and cook, stirring often, until blistered. Green beans should still have a slight crunch to them.
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.