Place a medium sauté pan over medium-high heat, add leeks, and add to pan. Season with salt, and sauté until leeks begin to soften, about 5 minutes, then reduce heat to medium-low. Stir in butter. Cover and cook, stirring occasionally, until leeks are very soft, about 20 minutes. Adjust salt and pepper to taste.
Preheat oven to 350 degrees. While leeks are cooking, spread bread cubes on baking sheet and bake until lightly golden brown, about 15 to 20 minutes. Turn halfway through baking. Transfer cubes to large bowl. Leave oven on.
Add sauteed leeks, thyme and chives to bowl of bread; toss well. In another large bowl, lightly whisk the egg and egg yolks, then whisk in milk or cream, a generous pinch of salt, pepper to taste and a pinch of nutmeg.
Sprinkle 1/4 cup shredded cheese in bottom of a buttered 9x12 baking pan. Spread 1/2 of bread mixture in pan, and sprinkle with another 1/4 cup cheese. Spread remaining bread mixture in pan, and sprinkle with remaining cheese. Pour in enough milk mixture to cover bread, and gently press on bread so milk soaks in. Let rest 15 minutes.
Add, if any, remaining milk mixture, letting some bread cubes protrude. Sprinkle with salt. Bake until pudding is set and top is brown and bubbling, about 30 to 35 minutes.