Preheat oven to 350 degrees with rack in upper third. Place potatoes in a medium saucepan and cover with 2 inches of water. Bring to a boil over high heat; reduce heat and simmer until potatoes are easily pierced with a knife, about 8 minutes. Drain potatoes, let cool, and cut into 1/4-inch-thick rounds.
Whisk together eggs, milk, cream, 1 teaspoon salt, 1/4 teaspoon pepper, and chives until thoroughly combined. Melt butter in an oven-safe 10-inch skillet over medium-high heat. Add leek and 1/4 teaspoon salt. Cook, stirring occasionally, until leek is softened and golden brown, about 5 minutes. Add thyme and cook, stirring gently, 30 seconds.
Reduce heat to medium. Pour egg mixture into skillet and stir once; cook, undisturbed, until edge is set, about 2 minutes. Scatter potatoes evenly over egg mixture. Transfer skillet to oven. Bake until edge is puffed and golden brown in places and center is set, about 20 minutes. Let cool slightly before serving.