Heat the butter in a small skillet over medium heat until melted.
Add the thinly sliced scallions (whites and greens) and cook, stirring occasionally, until soft and lightly browned, about 2–3 minutes.
Stir in the cooked white beans with their reserved broth. If the mixture looks thick, add 1/4 to 1/2 cup additional bean broth or vegetable broth so it resembles a thin soup.
Bring to a gentle boil, then reduce heat to a simmer and cook until the liquid has reduced slightly and the flavors meld, about 4–6 minutes; the result should remain a bit brothy.
Remove from the heat and whisk in the light miso and lemon juice until the miso is fully dissolved. Taste and adjust lemon or miso as desired.
Serve immediately with a generous crack of freshly ground black pepper.