First we will begin making the dough. Heat up milk in a small sauce pan until it begins to simmer. Remove from heat and add sugar and salt and stir. Let it cool down to lukewarm to the touch. Once milk is cool, add to a bowl of a stand mixer, along with the flour, sugar, melted butter, egg, egg yolks, and yeast. Attach the dough hook to the stand mixer and mix on low until dough forms. Increase speed to medium and knead dough until it's smooth and no longer sticks to the sides, about 10 minutes. If the dough is sticking for a while, add more dough until it only sticks to the bottom.
Reduce speed to medium low. Add butter to the dough, 1 tablespoon at a time until all the butter is absorbed. Do not add the next tablespoon until you don't see the previous tablespoon anymore. Once all the butter has been added, increase the speed and knead for 6 more minutes. Once you finish kneading, shape into a ball. Lightly grease another large bowl and place the ball of dough into bowl. Cover with a kitchen towel and let it rise for 1.5 hours.
Prepare two baking sheets by lining with parchment paper. Once dough has risen, punch it down with your fist and remove from the bowl. Lightly dust your counter with flour and place the dough on top. Form into a disk. Cut dough into 4 even slices. Cut each slice into 3 even pieces. You should have 12 even pieces. Roll each piece of dough into a ball and place on a baking sheet with parchment paper. Make sure to space the balls a couple inches apart and only 6 balls per baking sheet. Cover each baking sheet with a damp rag and let them rise until fluffy, about 45 minutes.
Make the lemon custard:
While the dough is rising, prepare the custard. Heat milk in a saucepan set over medium heat until steaming. In the meantime, whisk eggs, sugar, lemon zest, flour, corn starch, and salt in a medium bowl. It will be hard at first but eventually the mixture will turn smooth and light yellow.
Slowly whisk hot milk into the yolk mixture. Add back to the sauce pan used to warm the milk. Whisk the mixture as it heats over medium heat. Do this until the mixture thickens and reaches 170 degrees, about 3 minutes. Strain the mixture through a mesh sieve to remove lumps and lemon zest. Whisk butter into strained custard. Cover and refrigerate to help the mixture thicken.
Make the strawberries:
Prepare the strawberries. Hull and slicee the strawberries lengthwise. Add to a medium bowl with lemon juice and sugar. Stir and cover.
Make the buns:
Preheat oven to 350 degrees. Make sure both oven racks are placed in the center of the oven. Once the balls have puffed up, remove the rag that covers them. Grab a 1/3-cup measuring cup and dip into flour. Then use it to smash the center of each ball of dough to create a well in the center. Redip the cup in flour if it starts to stick to the dough. Fill each well with about 1 tablespoons of lemon custard and about 5 or 6 strawberry slices.
Brush the exposed edges of the buns with egg wash. To make the egg wash, whisk egg with 1 tablespoon of water. Sprinkle with pearl sugar.
Bake both sheets of buns in preheated oven until they are golden brown on the edges, about 18-20 minutes. Make sure to switch and rotate baking sheets halfway to ensure even baking.
Remove from oven and let cool for 15 minutes. Serve immediately or within 24 hours.