Put the potatoes in a large saucepan and cover with cold water by 1 inch. Add a generous pinch of salt, bring to a simmer and cook until the potatoes are easily pierced with a knife, about 40 minutes. Reserve 1/2 cup cooking water, then drain the potatoes and return to the saucepan.
Meanwhile, combine the olive oil, garlic, lemon zest, thyme and rosemary in a small saucepan over medium heat. Cook until the garlic starts to sizzle, about 30 seconds. Reduce the heat to medium low and cook until the garlic has softened and browned, 2 to 3 more minutes. Remove from the heat and discard the lemon zest and herb sprigs with a slotted spoon.
Add the garlic and oil to the saucepan with the potatoes along with 3/4 teaspoon salt and a few grinds of pepper. Lightly smash with a potato masher, gradually adding the reserved cooking water as needed, until combined. Transfer to a serving dish and drizzle with olive oil; top with the chives just before serving.