Start with browning the butter. Place the butter in a medium saucepan over medium heat. Let it melt completely, then continue cooking, swirling the pan occasionally, until it foams and the milk solids begin to brown on the bottom, about 5 to 8 minutes. Remove it from the heat immediately to prevent burning; pour it into a heatproof bowl or liquid measuring cup to cool.

In a medium saucepan, whisk together the flour and milk. Set over medium heat and cook, whisking constantly, until the mixture thickens and reaches at least 150°F on a kitchen thermometer, about 2 to 3 minutes.

Transfer the tangzhong to the bowl of a stand mixer fitted with the dough hook. Add the yeast, milk, sugar, 4 tablespoons of the brown butter (it will still be melted), egg, salt, and flour. Knead until the dough mostly pulls away from the sides of the bowl but still clings slightly to the bottom, appearing smooth and feeling tacky but not sticky, 20 to 30 minutes.

Cover the dough tightly with plastic wrap and place it in a warm spot to rise until doubled in size, about 1 hour.

Divide the dough into 9 equal pieces. Without flouring the surface, roll one piece into an 8x3-inch rectangle. Fold it in thirds lengthwise, rotate 90 degrees, and roll it flat again. Roll up into a cylinder, tuck the ends under to form a ball, and gently cup and drag it across the counter to create a smooth surface.

Arrange the balls in a parchment-lined 10-inch cast-iron skillet or baking dish. Cover tightly with plastic wrap and proof until nearly doubled in size, about 45 minutes.

Preheat the oven to 350ºF with a rack in the lower third position. Brush the tops with egg wash, then bake until deep golden brown and the internal temperature reaches at least 190ºF, 30 to 35 minutes.

Meanwhile, in a saucepan, fry the chopped sage in a dab of butter until crispy and fragrant, 1 to 2 minutes. In a medium bowl, using an electric mixer, whip together the fried sage, remaining brown butter, additional softened butter, and honey until very light and fluffy. Season generously with salt.

Brush some of the soft, whipped butter over the warm buns and sprinkle generously with flaky sea salt. Transfer the remaining butter to a serving bowl.