In a large pot, cover the potatoes with cold water and season them generously with kosher salt. Bring to a boil, then lower a simmer. Cook until the potatoes are almost tender but not completely cooked through, about 5 to 8 minutes. Using a slotted spoon, remove the potatoes to a large bowl. Drizzle with olive oil and toss to coat. Let cool.
Return the water to a boil and blanch the string beans for 3 minutes, then transfer to an ice bath. This helps to retain the vibrant color and crispness. Strain and place in a separate bowl. Drizzle with olive oil and toss to coat. Let cool.
Whisk together the vinaigrette ingredients in a separate bowl.
Heat the grill to medium-high. Using a grill tray or basket, arrange the potatoes in a single layer, cut side down. Cook until nicely browned, about 3 to 5 minutes.
Remove the potatoes, placing them back in the same bowl they were in.
Place potatoes and green beans in a large serving bowl. Toss with the vinaigrette.
Top with torn basil and Nasturtium flowers, if available.