Prepare milk tea cream filling. Combine heavy cream with tea leaves in a medium saucepot. Bring to a simmer over medium heat, reduce heat to low and continue to steep tea for 5 minutes. Remove from heat and let sit for 1 hour. Strain cream, discarding tea. Chill in the fridge.
Make bread dough. Combine milk with tea leaves in a medium saucepot. Bring to a simmer over medium heat. Reduce heat to low and steep tea for 5 minutes. Remove from heat and let sit for 10 minutes. Strain milk, discarding tea leaves. Measure ¾ cup of the infused milk, reserving any extra for another use. If you do not have ¾ cup, add enough plain milk to yield ¾ cup. Let milk cool slightly if it's above 100 degrees F.
Combine tea infused milk with active dry yeast and 1 teaspoon sugar, stirring gently to combine. Let sit for 5 minutes or until yeast is activated and foamy.
In the bowl of a stand mixer, add flour, ¼ cup sugar, salt, egg, softened butter, and yeast mixture. Knead dough until smooth, about 7-8 minutes. Transfer dough to a bowl, cover with plastic wrap and let proof in a warm area for 1 hour or until doubled in size.
Line two baking sheet trays with parchment paper. Punch down dough and divide it into 12 equal portions. Roll each portion into a ball and place on prepared baking sheet, 6 buns per sheet. Make sure to evenly space the dough to give them enough room to expand. Cover the buns with a clean kitchen towel and let proof in a warm area for 1 hour or until doubled in size.
Meanwhile, prepare red bean filling. Rinse dry adzuki beans with water and drain. Add beans and water in the instant pot and pressure cook for 20 minutes. Let sit for 25 minutes and release pressure if needed. Drain beans, reserving ¼ cup of the cooking liquid. Puree beans in a food processor with ¼ cup cooking liquid until smooth.
Melt butter in a large nonstick saute pan over medium heat. Add red bean puree, stirring until incorporated with the butter. Reduce heat to low and add half of the brown sugar, stirring until sugar has melted. Add remaining sugar and continue to heat until sugar is fully absorbed, about 2 minutes. Remove from heat and transfer to a bowl. Cover with a damp paper towel and let cool.
Preheat oven to 350 degrees F.
Brush buns with egg wash. Bake for 15-20 minutes or until golden brown, rotating the trays halfway. Remove from the oven and transfer buns to a wire rack to cool completely.
While buns are cooling, make milk tea whipped cream. Add tea infused heavy cream, powdered sugar, and vanilla in the bowl of a stand mixer and whip with the whisk attachment until stiff peaks form. Alternatively, whip by hand. Transfer cream filling to a piping bag with an open star piping tip (tip is optional).
Cut buns in half and set aside the top bun. Spread 2 tablespoons red bean paste on the bottom bun and pipe whipped cream on top. Top with remaining bun and serve immediately.