Brush a 9-inch square baking dish with olive oil. Sprinkle the baking dish with lemon zest and set aside.
In a medium bowl, whisk the olive oil, mustard, salt, and pepper with a splash of the milk. Whisk in remaining milk and the eggs.
Put the bread in the prepared baking dish and top with the spinach and half of the feta. Gently toss with your hands until combined. Drizzle the egg mixture over the bread and sprinkle with the remaining feta. Cover and refrigerate overnight.
Preheat the oven to 350 degrees with a rack in the top third of the oven.
Bake the strata, uncovered, for 45 to 55 minutes, until the egg is set in the middle and the edges are browned. Brown the top under the broiler, if desired. Serve warm, drizzled with a bit of olive oil and sprinkled with the fresh oregano.