Leek Bread Pudding

Adapted from Ad Hoc at Home, by Thomas Keller
Serves 12 as a side dish, 6-8 as a main course
2 cups 1/2" thick slices leeks (white and light green parts only)
Kosher salt
1 tablespoon canola or vegetable oil
4 tablespoons (2 oz.) unsalted butter
Freshly ground black pepper
12 cups 1" thick cubes crustless Brioche or Pullman sandwich loaf
1 tablespoon finely chopped chives
1 teaspoon thyme leaves
3 large eggs
3 cups whole milk
3 cups heavy cream
Freshly grated nutmeg
1 cup shredded Comte or Emmentaler
Preheat the oven to 350 degrees.
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 saute pan over medium-high heat, add the canola oil and butter.  When the foaming subsides, add the leeks and salt to taste, and cook, stirring often, until they begin to soften, about 5 minutes.  [Keller tells you to make a parchment lid at this point but I truly don't see why you couldn't just put a real lid on the pot leaving it partially open].  Continue to cook, stirring often, until the leeks are very soft, 30 minutes [mine did not take this long].
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 9x13" 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 the 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.  Let stand 10 minutes before serving.