Preheat oven to 400 degrees F. For the leeks: Melt butter in large skillet over medium-high heat. Add leeks and salt and sauté for 4 minutes. For the bread cubes: In a large bowl, toss the bread cubes with melted butter, thyme, and salt. Spread out on a large baking sheet and bake until golden and slightly crunchy, 10 minutes.
Add two tablespoons of water to the leeks and cover the skillet to steam, 10 minutes; remove leeks to a baking sheet to cool for 2 minutes. For the custard base: In a large bowl, whisk egg, egg yolks, heavy cream, milk, half the Parmesan, and salt.
Assemble the bread pudding: Reduce oven temperature to 325 degrees F. In a large mixing bowl or pot, layer the croutons, leeks, and custard base for even mixing. Add remaining Parmesan and mix thoroughly to combine. Butter a casserole dish, add the bread pudding and finish with more Parmesan. Bake 25—30 minutes.
Remove casserole dish from the oven when the custard has set and the top is golden brown; serve.