In a large pot add potatoes, cover with water, sprinkle in a little kosher salt, and throw in thyme sprigs and bay leaf.
Bring to a boil over high heat and cook until the potatoes are fork tender, about 15 minutes.
While the potatoes are cooking whisk together red wine vinegar, grated garlic, anchovy paste, minced capers, and Dijon mustard.
Slowly drizzle in olive oil while whisking until you're left with a smooth mixture.
Taste and season with the desired amount of kosher salt and black pepper - set aside.
When the potatoes are fork-tender, drain and add to a large bowl.
Drizzle half the dressing over the potatoes and stir to combine. Set aside while you prepare the remaining ingredients.
Add water back to the pot you used to boil your potatoes and bring it to a boil.
Add in your beans and cook for just 4 minutes or just until the beans are slightly soft.
Strain the beans and add to a bowl of iced water. This will help stop the cooking so your beans stay bright green and a little crunchy.
Remove beans from the cold water and add to the potatoes.
Drizzle with remaining dressing and toss to combine.
Before serving add the arugula and toss to combine until everything is coated in dressing. If the salad looks dry drizzle in a little more olive oil.
To serve, divide the salad into 4 equal portions, and serve each portion with a hard or soft-boiled egg. Sprinkle the eggs with a little additional kosher salt and black pepper.