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1
In a small bowl, mix together the mayonnaise, miso, mustard, paprika, lemon zest, lemon juice, and black pepper. Taste for seasoning and balance, adding a bit more of any of the ingredients to your liking.
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2
In a large pot, cover the potatoes with cold water and season generously with kosher salt. Bring to a boil, then lower heat to maintain a simmer. Cook until the potatoes are almost tender but not completely cooked through, about 5 to 8 minutes. (The total cooking time will depend on the potatoes' freshness, so check their doneness a few times.) Using a slotted spoon, remove the potatoes to a large bowl; dry them if needed. Immediately toss the potatoes with 1/4 cup of the mayo sauce.
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3
Return the water to a boil, then add the green beans. Cook until they're crisp-tender and just shy of done, about 2 to 3 minutes. Drain, then add to a separate bowl. Immediately toss the green beans with 2 tablespoons of the mayo sauce.
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4
Heat the grill to medium-high and brush your grates clean. Using a grill tray or basket (brushed with oil), arrange the potatoes in a single, even layer -- cut side down. Cook until nicely browned, about 3 to 5 minutes, then flip and cook a few minutes more. Remove potatoes, placing them back in the same bowl they were in.
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5
Repeat process with the green beans, cooking until they're nicely charred on the first side, about 2 to 3 minutes, then flipping and cooking a few minutes more until browned all over. Add green beans to the same bowl with the potatoes. Adjust seasoning to taste.
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6
Place potatoes and green beans on a large serving platter or dish, then top with basil. Spoon a little mayo across the top before serving, if desired. Serve warm or at room temperature, with the rest of the mayo sauce on the side.
