Start with the tangzhong by placing bread flour and milk in a saucepan. Heat over medium-high heat and whisk until it comes together, about one minute. Set aside and allow to cool.
Combine the yeast and milk and let it rest for five minutes, allowing the yeast to become active. Combine the bread flour, cricket flour, fennel, salt, and sugar in a large bowl of a stand mixer. Add the tangzhong, milk, and egg to the bowl and mix until it just comes together.
While the dough begins to knead, add the butter one tablespoon at a time. Knead the dough for five minutes. If the dough is too sticky, add one tablespoon of flour at a time. If it is too dry, add one tablespoon of milk at a time.
Transfer to an oiled bowl, cover, and let rest for one hour or until it has doubled in size. Once doubled, punch down and knead. Prepare a baking sheet by placing parchment paper down.
Divide the dough into six equal portions. Shape into round balls and place on the baking sheet. Cover with plastic wrap and let rest for 30 to 60 minutes, or until doubled in size.
Preheat an oven to 375°F. In a small bowl, whisk an egg and two tablespoons of water. Remove the plastic wrap from the baking sheet and brush each of the buns with egg wash. Place in the oven and bake for 16 to 19 minutes or until golden brown.