Adjust oven rack to middle position and preheat oven to 375°F (190°C). Place potatoes in a microwave-safe bowl; cover with plastic wrap, and cut a small slit in plastic wrap to allow steam to escape. Microwave on high, stirring and turning potatoes halfway through cooking, until fork-tender, 8 to 9 minutes. Transfer potatoes to a rimmed baking sheet, spread in a single layer, and let cool, about 10 minutes.

While potatoes are cooking, in a large skillet, melt butter over medium heat. Add leeks and ½ teaspoon kosher salt; cook, stirring occasionally, until lightly browned, softened, and moisture has cooked off, 6 to 7 minutes. Remove from heat; let cool slightly, about 10 minutes. Transfer 2 tablespoons of cooked leeks to a small bowl and set aside.

Coat a 13- x 9-inch baking dish with butter. Arrange half of potatoes in an even layer on bottom of prepared baking dish. Top with half of ham, half of cooked leeks from skillet, and sprinkle with 1 cup cheese. Repeat layers once more with remaining potatoes, remaining ham, remaining leeks, and 1 cup of the cheese.

In a large bowl, whisk together eggs, half-and-half, mustard, thyme, 1 tablespoon chives, pepper, and remaining 1 teaspoon kosher salt until fully combined. Pour custard evenly over mixture in baking dish. Top with remaining 1 cup cheese.

Bake until cooked through, center is set and lightly puffed, top is lightly browned in spots, and the edges are golden and slightly crisp, 35 to 40 minutes.

Let rest 15 minutes before slicing. Garnish with remaining 1 tablespoon chopped chives, reserved leeks, and more thyme leaves, if desired. Slice and serve.