Pour warm milk into the mixing bowl of a stand mixer. Dissolve yeast and sugar in milk. Let sit for 5 minutes to activate yeast. Whisk in egg. Attach dough hook to stand mixer.
In a large bowl, whisk together all-purpose flour, bread flour, and salt. Add half of flour mixture to mixing bowl. Mix on low speed until dough starts to form together. Continue running the mixer of low speed and add softened butter one tablespoons at a time. Once all the butter has been incorporated, add remaining half of flour.
Increase to medium speed and knead bread for 2 minutes until dough is smooth, elastic, and does not stick to the sides of the mixing bowl. Transfer dough to a greased large bowl. Cover and allow dough to rest for 1 hour, until doubled in volume.
Punch down dough. Transfer to a clean working surface. Roll dough into a 20 x 14-inch rectangle.
Filling: Mix together softened butter, sesame oil, and salt. The butter mixture should be creamy like mayonnaise. Spread filling mixture over rolled dough. Top with chopped scallions and shredded cheddar.
Starting at the longer side, roll dough into a log. Slice the log into 12 equal parts. Place cut rolls in a greased 9 x 12-inch baking pan. Cover with plastic wrap and let rolls rest for 30-45 minutes at room temperature to allow rolls to rise.
Preheat oven to 350°F.
Egg wash: In a small bowl, whisk together egg, water, and salt. Lightly brush egg wash mixture over risen rolls. Sprinkle sesame seeds over rolls.
Bake rolls for 30-35 minutes until golden brown on top and the internal temperature of the rolls register at 185°F. Allow the rolls to rest for 5 minutes to slightly cool before serving.