Melt butter in a saucepan over medium heat. Add milk & heat until lukewarm, but not hot. Stir in yeast. Allow yeast mixture to proof for about 10 minutes.
In a large mixing bowl, combine flour, sugar & salt; add both eggs & combine. Add the yeast/milk/butter mixture. Continue to mix until the dough forms a ball & there is no dough sticking to the sides of the bowl.
Turn dough out onto a floured surface & knead well until dough is smooth (about 10 minutes). Form dough into a ball. Grease mixing bowl with butter. Place dough in the bowl, cover & allow to rise until its roughly doubled in size (about 1 hour).
Peel, halve & slice persimmons into 1/4-inch thick slices. Place in a shallow dish with sugar & cardamom; toss gently to evenly coat slices. Set aside.
Cream Cheese Filling
In a bowl, whisk together softened cream cheese with milk. Gradually add in the powdered sugar, beating until mixture is smooth. Add lemon juice & set aside.
Once the dough has risen, turn it out onto a floured surface & punch down. Divide it into 12 equal parts. Roll the dough balls into discs 4-5-inches in diameter & about 1/4-inch thick. Place the discs onto a parchment lined baking sheet. Cover with a buttered piece of plastic wrap. & allow the dough to rise again in a draft-free place for about 25 minutes.
Preheat oven to 350 F.
When dough has risen, using your fingers, press the center of each disc down, leaving about a 1/2-inch rim. If necessary, you can dip your fingers in egg wash to keep the dough from sticking during this process.
Assembly & Baking
Spread about a teaspoon of the cream cheese filling in a thin layer in the depression of each disc. Press some persimmon slices in the center over the cream cheese. Brush the outer edges of the discs with egg wash mixture. Place buns in the oven & bake for about 20-25 minutes or until golden. Remove from oven & transfer to a wire rack, keeping them on the parchment paper.
Combine the apricot jam & warm water. Microwave to thin the jam to a liquid consistency. Brush buns LIGHTLY with glaze & garnish with almond slices & pearl sugar.