Preheat the oven to 350°. Lightly oil a medium baking dish. Spread the bread cubes on a baking sheet and bake for about 10 minutes, until dry but not browned. Let cool, then transfer to a large bowl. Stir in the spinach and 1/4 cup of the feta.
In another bowl, whisk 2 tablespoons of the olive oil with the mustard, lemon zest and lemon juice. Add the eggs and beat until blended. Add the milk and season with 1 teaspoon of salt and 1/2 teaspoon of pepper. Pour the egg mixture over the bread cubes and stir until they are evenly moistened. Transfer the bread mixture to the baking dish and let stand at room temperature for 2 hours or refrigerate overnight.
Sprinkle the remaining feta on the bread pudding and bake in the center of the oven until risen and set, about 40 minutes. Turn on the broiler. Drizzle with the remaining 1 tablespoon of oil and broil until the bread pudding is golden and crispy on top, about 2 minutes. Scatter the oregano on top, cut into squares and serve.