First, make the filling: In a medium bowl, add the beans and enough cold water to cover by 3 inches. Cover and set aside at room temperature to soak for at least 6 hours or overnight.
Drain the beans, discarding the soaking liquid. In a medium pot, add the beans and enough cold water to cover by 1½ inches. Bring to a boil over medium-high heat, then immediately remove from the heat and drain, discarding the cooking liquid. Repeat this blanching process twice more, using fresh water and discarding the cooking liquid each time.
Return the beans to the pot and cover with enough cold water to cover by ½ inch. Bring to a boil over high heat, then lower the heat to maintain a simmer. Cover with a lid and cook, stirring occasionally, until the beans are cooked through and can be mashed easily with a wooden spoon, 35–40 minutes. Stir in both sugars and the salt until completely dissolved, about 1 minute more. Transfer to a food processor and purée until completely smooth. Transfer to a heat-resistant container, cover, and refrigerate until completely chilled.
Make the mochi: In a small pot over medium-low heat, add the shiratamako, granulated sugar, and ½ cup plus 1 Tbsp. (130 g) cold water. Cook, stirring constantly, until the mixture forms a paste, and turns translucent and sticky, 2–3 minutes.
Dust a clean work surface liberally with jyoushinko. Place a fine-mesh strainer over the pot of shiratamako paste, then sift the jyoushinko and matcha into the pot. Using a wooden spoon, fold in the jyoushinko mixture until completely incorporated, then turn the paste out onto the prepared work surface. Dust your hands with more jyoushinko. Shape the dough into a tidy square, then roll it out to a very thin, even 11-inch square. Using a biscuit cutter, bicycle cutter, or pizza wheel, cut the dough into sixteen 2¾-inch squares.
Using a teaspoon, scoop a dollop of filling into the center of each square. Lightly brush the edges of each square with cold water, then fold into triangle-shaped dumplings, pinching the edges gently to seal. Dust with kinako and garnish with flowers before serving (if desired).