Preheat your oven to 350°F (175°C) and lightly grease a small baking dish to prevent sticking.
In a mixing bowl, whisk together the eggs and sugar until the mixture becomes smooth and begins to lighten in color, with a gentle foaming sound.
Add the finely grated lemon zest to the egg mixture, whisking to distribute the bright citrus aroma evenly.
Slowly pour in the strained lemon juice while whisking constantly, watching the mixture turn slightly pale and silky.
Warm the milk until just steaming, then gradually whisk it into the lemon egg mixture, helping it stay smooth and cohesive.
Pour the smooth custard mixture into the prepared baking dish, smoothing the top with a spatula for an even surface.
Place the baking dish into a larger baking pan and add hot water to the outer pan until it reaches halfway up the sides of the custard dish, creating a gentle water bath.
Bake in the preheated oven for about 30 minutes, until the custard is just set around the edges but still slightly wobbly in the center and the top begins to turn a light golden.
Remove the dish from the oven and water bath, then let it cool slightly. The custard will firm up as it cools, developing a silky texture.
Serve the Baked Lemon Pudding warm or chilled, enjoying the fragrant, creamy custard with a hint of bright lemon zest on top.