Cook pork with 1/4 cup water in a large, wide, heavy pot (stainless steel or enameled iron) over high heat, stirring occasionally, until all liquid has evaporated, 20 to 30 minutes.
Add lard and reduce heat to low. Move pork chunks as needed to keep them submerged. Cook pork until starting to soften on the edges but still firm in the center, about 1 hour.
Whirl garlic and salt together in a food processor to make a paste. Stir garlic paste into lard.
Cook pork until fork tender, about 1 hour more. Lift pork into a strainer set over a bowl. When cool, break into smaller pieces.
Spoon about 1 tbsp. melted lard into a large frying pan. Lay a tortilla in pan, sprinkle evenly with about 1/4 cup cheese, and set another tortilla on top to create a quesadilla. Cook over medium heat, turning once, until cheese has melted, 4 minutes. Repeat with remaining tortillas.
Fill quesadilla with about 1/2 cup carnitas, beans, and salsa.
*Find lard (manteca in Spanish) at Latino markets, or order it from a butcher. Freshly rendered tastes best.
Note: Nutritional analysis is per taco without salsa.