Grease a 2-quart baking dish with nonstick spray. Place soda bread pieces in the dish.
Combine hot water and instant espresso in a large bowl; whisk until the espresso blends into the water. Add brown sugar and cooled coffee and whisk until well combined. Mix in milk and eggs until well blended and smooth. Add vanilla and salt, whisking again until custard mixture is well blended.
Pour custard mixture over the bread cubes, cover with foil, and refrigerate or let stand at room temperature until custard has soaked into the bread, about 1 hour.
Preheat the oven to 350 degrees F (175 degrees C).
Bake the covered pudding until custard is fully set in the middle, 40 to 45 minutes. Remove foil and sprinkle with coarse sugar. Bake until top is lightly browned, about 10 more minutes. Serve warm.
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