Boil 1" water in a 4-qt. saucepan fitted with a steamer insert. Steam potatoes,
covered, adding more boiling water as needed, until tender, 1 hour. Let cool, then peel and thinly slice.
Heat oven broiler. Heat oil in an ovenproof 12" nonstick skillet over medium-high heat. Cook garlic, pepper, and onion until soft, 3–4 minutes. Add spinach; cook until wilted, about 1 minute. Stir in reserved potatoes, the butter, salt, and pepper. Stir in half the basil and the eggs and reduce heat to medium; cook until golden on the bottom, 8–10 minutes. Broil until set and golden on top, about 3 minutes. Garnish with remaining basil.