Place the potatoes in a large pot, cover them with cold water, then sprinkle with 1/2 teaspoon kosher salt. Bring to a boil, reduce heat to a simmer, then cook until fork-tender, about 10-15 minutes. With a slotted spoon, gently remove the potatoes from the water and place them in a large bowl. Set aside.
Return the water to a gentle boil, then add the green beans to the pot. Blanche just until crisp-tender, about 3 minutes. Drain into a colander, then quickly rinse the beans with cold water to stop their cooking.
In the bowl of a food processor fitted with a steel blade, add the basil, arugula, 1/3 cup almonds, Parmesan, garlic clove, 1 tablespoon lemon juice, and 1/4 teaspoon salt. Pulse the ingredients until evenly ground. With the machine running, slowly pour the olive oil into the feed tube and blend until smooth.
While the potatoes are still warm, pour the pesto over the top, then toss gently to combine, being careful to break up the potatoes as little as possible.
Add the green beans to the bowl with the potatoes, then add the peas and the remaining 1/4 teaspoon kosher salt. Stir all of the gently to combine.
Squeeze a little fresh lemon juice over the top, then sprinkle with 2 tablespoons chopped toasted almonds. Serve warm, at room temperature, or chilled.
TO STORE: Leftover pesto potato salad will keep well in the refrigerator for 1 full day so it can easily be made a day ahead as well. Note that by days 2 and 3, it will still be edible but will lose some vibrancy and crispness.