Put the leek rounds in a large bowl of tepid water and swish so that any dirt falls to the bottom of the bowl. Set a medium sauté pan over medium-high heat, lift the leeks from the water, drain, and add them to the pan. Season with salt and cook, stirring often, for about 5 minutes. As the leeks begin to soften, lower the heat to medium-low. The leeks will release liquid. Stir in the butter to emulsify, and season with pepper to taste. Cover the pan with a parchment lid, and cook, stirring every 10 minutes, until the leeks are very soft, 30 to 35 minutes. If at any point the butter breaks or looks oily, stir in about a tablespoon of water to re-emulsify the sauce.
Remove and discard the parchment lid.
Meanwhile, spread the bread cubes on a baking sheet and toast in the oven for about 20 minutes, rotating the pan about halfway through, until dry and pale gold.
Transfer to a large bowl. Leave the oven on.
Add the leeks to the bread and toss well, then add the chives and thyme.
Lightly whisk the eggs in another large bowl.
Whisk in the milk, cream, a generous pinch of salt, pepper to taste, and a pinch of nutmeg.
Sprinkle 1/4 cup of the cheese in the bottom of a 9-by-13-inch baking pan.
Spread half the leeks and croutons in the pan and sprinkle with another 1/4 cup cheese. Scatter the remaining leeks and croutons over and top with another 1/4 cup cheese.
Pour in enough of the custard mixture to cover the bread and press gently on the bread so it soaks in the milk.
Let soak for about 15 minutes.
Add the remaining custard, allowing some of the soaked cubes of bread to protrude.
Sprinkle the remaining 1/4 cup cheese on top and sprinkle with salt.
Bake for 1 1/2 hours, or until the pudding feels set and the top is brown and bubbling.
