Preheat your oven to 350°F (175°C). Butter a 9x13-inch baking dish generously and set aside.
Spread the bread cubes on a baking sheet and toast in the oven for 10 minutes until slightly dried but not browned. Transfer to the prepared baking dish.
Sift the matcha powder with 2 tablespoons of the sugar in a small bowl to remove any lumps. Set aside.
In a large mixing bowl, whisk together the eggs, remaining sugar, and salt until smooth and slightly thickened, about 1 minute.
Add the warmed milk, heavy cream, vanilla extract, and the matcha-sugar mixture to the egg mixture. Whisk vigorously until the matcha is fully dissolved and the custard is uniformly green with no streaks.
Pour the matcha custard evenly over the toasted bread cubes, pressing down gently with a spatula to ensure all pieces are submerged. Let stand for 20-30 minutes at room temperature, pressing down occasionally.
Bake uncovered for 40-45 minutes until the top is golden and slightly puffed, and a knife inserted in the center comes out with just a bit of creamy custard clinging to it.
Remove from the oven and let cool for 10 minutes before serving. The pudding will set further as it cools but is best enjoyed warm.
