Make the bagna cauda: Combine the olive oil, butter, and sliced garlic in a small pot over low heat. Cook until the garlic is completely soft and tender, 10-15 minutes, lowering the heat even further as needed to maintain a subtle simmer and prevent the garlic from browning.
Add the anchovies (don't worry about chopping them—they're going to dissolve in the hot fat) and continue to simmer, stirring occasionally, until the anchovies are completely dissolved and have begun to fry lightly in the fat—they'll start to smell a little nutty and browned—15-20 minutes.
Add the scallion greens and grated garlic. Continue to cook until the scallions are wilted and the garlic is fragrant, another minute or so. Remove from the heat and set aside.
Make the beans: Combine the beans and bagna cauda in a medium pot over medium-low heat. Add the bay leaf (if using) and let simmer together for 20-30 minutes. Taste a bean or two—they should be garlicky and buttery, herby and salty with anchovy. Adjust with salt, pepper, and maybe another few anchovies if you like (otherwise, save them for topping at the end).
To serve, pour into a large serving bowl and squeeze lemon over everything. Top with more anchovies, if you like, and maybe some parsley.
