In the bowl of a stand mixer fitted with the paddle attachment, whisk together warm milk, sugar, and yeast by hand. Let stand until foamy, about 10 minutes.
Add flour, Tangzhong, egg, and kosher salt to yeast mixture, and beat at very low speed until combined, about 30 seconds. Switch to the dough hook attachment. Beat at low speed until smooth and elastic, about 6 minutes. Add softened butter, 1 tablespoon (14 grams) at a time, beating until combined after each addition, about 3 minutes total. Beat until dough is smooth and elastic, about 1 minute.
Turn out dough onto a lightly floured surface, and shape into a smooth round. Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 45 minutes to 1 hour.
Spray 2 (5½x3-inch) loaf pans with cooking spray. Line pans with parchment paper, letting excess extend over sides of pans.
Punch down dough, and turn out onto a lightly floured surface. Roll dough into a 12-inch square, about ½ inch thick. Spread cream cheese onto dough, and sprinkle with fontina, mozzarella, and provolone. Cut dough into 2-inch squares. Place pieces, cut sides up, in each prepared pan, squishing dough as needed. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, 20 to 30 minutes.
Preheat oven to 350°F (180°C).
Bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), about 35 minutes, covering with foil after 20 minutes of baking to prevent excess browning. Let cool in pan for 10 minutes. Using excess parchment as handles, remove from pans. Brush with melted butter, and sprinkle with sea salt. Serve warm.
Notes
To make one 8x4.5-inch loaf, cut dough into 4-inch squares instead of 2-inch squares, and bake for about 40 minutes.
Recipe by Bake from Scratch at https://bakefromscratch.com/small-batch-four-cheese-pull-apart-milk-bread/