In a stand mixer, combine bread flour, instant yeast, sugar, salt, water, and egg. Mix until cohesive, about 10 minutes. Gradually add softened butter, mixing until smooth and elastic, about 5 minutes.

Turn the dough out onto a floured surface and divide into 9 equal pieces. Roll each into a ball and place on a parchment-lined baking sheet. Cover with a towel and let rest for 20 minutes.

Boil potatoes until tender, about 15-20 minutes, then mash them. Boil eggs for 10 minutes, cool, and mash. Sauté onion, green onion, carrot, and ham; add soy sauce. Mix with mashed potatoes and eggs.

Flatten each dough ball into a circle. Place 2-3 tablespoons of filling in the center, fold the dough over, and pinch edges to seal. Flatten gently to form a croquette shape.

Mix water, flour, salt, and baking soda in a dish. Dip each croquette, then roll in breadcrumbs. Let them rest for 15 minutes.

Heat oil in a deep pan to 340°F (170°C). Fry croquettes in batches for about 2 minutes per side or until golden brown. Drain on paper towels.