In a small saucepan, whisk the sticky rice flour and water until smooth, then cook it over medium-high heat, stirring constantly. It will appear lumpy at first, then eventually it becomes a big glob of sticky goo. When it does, turn off the heat and let it cool completely.
In the bowl of a stand mixer with a dough hook attachment, combine the sticky rice goo, bread flour, egg and egg white, sugar, yeast and salt. Knead on low speed until the mixture comes together into a sticky dough (the dough may seem dry at first, but do not add water.
Add the butter, scraping the sides and bottom of bowl to help the butter incorporate evenly, and knead for 7–8 minutes, until the dough is extremely smooth and elastic. The dough should be sticky, but only sticking to the bottom of the bowlas the machine is running, leaving a thin opaque film on the sides of the bowl. It is too wet—sticking everywhere as the machine is running—add 1 tablespoon of bread flour at a time and knead again until it looks right.
Cover the bowl with plastic wrap and let the dough rise at room temperature until doubled, 2 to 3 hours.