Make the dough. Combine warm milk with honey and active dry yeast. Let sit for 5 minutes or until the yeast is activated and the mixture is foamy.
Combine yeast mixture with egg, bread flour, and salt and mix until a rough dough comes together. Add butter and knead for 5-8 minutes or until smooth and elastic. Add 1-2 tablespoons flour if the dough is too sticky. Transfer the dough to a bowl, cover with plastic, and let proof for 1 hour or until doubled in size.
Meanwhile, make the filling. Place potatoes in a medium pot and cover with water. Bring to a boil over medium-high heat and cook until the potatoes are fork-tender. Drain and mash the potatoes.
Heat 1 tablespoon of oil in a large saute pan. Add the ground beef and saute until browned, breaking up the meat into small pieces. Add chopped carrot, onion, garlic, and scallion, and saute for 3-4 minutes. Add chicken bouillion powder and water and bring to a simmer. Continue to cook until all of the liquid has evaporated, stirring occasionally. Season with salt and pepper and remove from heat.
Combine the cooked beef and vegetables with potatoes, and mayo. Season with salt and pepper and mix the filling until well combined. Let cool completely.
Punch down the dough and divide it into 8 equal portions. Roll each portion into a ball, cover loosely with a kitchen towel and let rest for 15 minutes.
Take one of the dough balls and roll it out into a 6-inch circle. Scoop ⅓ cup of the filling into the center of the dough and gather the edges to seal. Flatten slightly to shape it into a patty and dip it into the beaten egg. Coat the croquette bread in panko, coating both sides. Place on a parchment-lined baking sheet and cover. Repeat with the remaining dough. Lightly cover the buns with a clean kitchen towel and proof the bread for 45 minutes.
Preheat oven to 400 degrees F.
Generously spray the croquette bread with cooking spray. Bake for 15-20 minutes or until golden brown. Remove from the oven, let cool slightly, and serve warm.