Combine the milk and vanilla bean in a 3-quart saucier. Heat over medium heat until steaming, then remove from heat and steep for 30 minutes. Alternatively, refrigerate up to 24 hours for a deeper vanilla flavor.

Whisk together sugar and cornstarch in a medium bowl. Add the eggs and mix until smooth.

Reheat the milk until steaming, then slowly whisk ½ cup of hot milk into the egg mixture to temper it. Gradually whisk in the rest of the milk.

Return the mixture to the saucier over medium heat. Cook, whisking constantly, until it thickens into a custard (about 3 minutes). Continue whisking for 2 more minutes to neutralize the starch-dissolving enzymes in the yolks.

Transfer to a 7x11-inch baking dish, press plastic wrap against the surface, and refrigerate until completely cooled, about 1 hour.

In a stand mixer fitted with a paddle attachment, beat cream cheese and butter on medium speed until light and fluffy, about 5 minutes.

Whisk the cooled custard until smooth. Gradually add it to the cream cheese mixture, beating well after each addition. Switch to the whisk attachment.

Whip the frosting on medium speed, adding lemon juice a little at a time, until fully incorporated. Whisk for a few seconds more to ensure smoothness.

Use immediately or let it stand at room temperature for a few hours. Refrigerate leftovers in an airtight container for up to 3 weeks or freeze for up to 3 months. Thaw and rewhip before using.