Trim most of the fat cap off the pork, leaving some intact to render in the pot. Cut it into two or three large pieces, season on all sides with salt and pepper. Set aside.
Cut the root ends and the dark green parts off the leeks, then slice in half. Run them under cold water, separating the layers to make sure all sand is washed off. Use a good chef's knife to cut into ½ inch slices. Peel the shallots and slice thinly. I like to leave them in slices because it's easier for prep and then you get carmelized shallot slices throughout the stew, creating wonderful flavor and texture.
Grate the garlic on a microplane grater, chop the sage finely. In a large 6-quart Dutch oven over medium-high heat, sear the pork pieces on all sides until crispy and golden brown. Set aside. Add shallots and leeks to the rendered fat in the Dutch oven, season with a pinch of salt, stir to scrape up any brown bits on the bottom of the pot. Cook over medium to medium-high heat until they are meltingly soft and starting to caramelize, 8-10 minutes.
Add the grated garlic and chopped sage to the pot, stirring constantly for about 1 minute until fragrant.
Add the whole bottle of wine to the pan, stirring constantly, season with 1 teaspoon of salt. Add all three cans of beans along with their liquid and the cup of water and stir to combine.
Add the pork and its resting juices back into the pan and bring to a simmer over medium heat. Once simmering, turn heat to low, cover and cook for 2 - 2 ½ hours, until pork is falling apart.
Remove pork from the pot and continue to simmer the stew uncovered until it thickens to your desired consistency, 15-30 minutes.
Using your hands (CAREFULLY, it is still VERY HOT) or two forks, shred the pork and add it back into the stew. Cook for about 5 minutes until the pork is heated through, adjust seasoning if needed, then serve.