Preheat the oven to 425 degrees F and set a wire rack on a rimmed baking sheet. Bring a medium saucepan of salted water to a boil. Add 2 cups basil; cook 30 seconds, then drain and rinse under cold water. Transfer to a food processor and puree with the butter, garlic, lemon zest and 1/2 teaspoon salt until smooth. Set aside.
Spatchcock the chicken: Cut along each side of the backbone with kitchen shears; discard the backbone. Open the chicken like a book; flip over. Press down on the backbone to flatten. Gently loosen the skin from the breasts and thighs using your fingers. Transfer half of the basil butter to a bowl and rub all over the chicken and under the skin; season with salt and pepper. Transfer skin-side up to the rack. Roast until the skin is browned and crisp and a thermometer inserted into a thigh registers 165 degrees F, about 45 minutes. Transfer to a cutting board and let rest 15 minutes.
Meanwhile, fill a medium saucepan with water; season with salt and add the potatoes. Bring to a boil, then reduce the heat and simmer until the potatoes are tender, about 12 minutes. Drain and transfer to a large bowl; toss with half of the remaining basil butter and season with salt and pepper.
Whisk the vinegar and mustard in a large bowl. Gradually whisk in the olive oil. Add the frisee, plums and the remaining 1 cup basil. Season with salt and pepper; gently toss to coat. Spread the remaining basil butter over the chicken; carve. Serve with the potatoes and salad.