Add the bacon grease and onions to a pan over medium heat. Cook for around 5-6 minutes, stirring frequently, then turn heat to medium-low.
Continue cooking for another 10 minutes, until onions are completely softened. Add the garlic at the 9-minute mark.
Add the beef cooking stock and beans, and increase the heat to medium. Add Mexican oregano, cumin, chili powder, cayenne pepper, and lime juice.
Simmer for 8-10 minutes, or until liquid is nearly evaporated. Reduce heat to low, then mash beans until desired consistency is reached. You can also use a stick blender if you prefer smooth, creamy refried beans.
Depending on how wet or dry you prefer your refried beans, you may want to add more beef cooking stock at the end to achieve the desired consistency. The beans should be pretty dry at this point. My preference is to slowly add a few splashes of beef cooking stock at a time until I reach a consistency I like.
Taste and add salt, if necessary. Remove from heat and garnish with crumbly cheese and sliced green onions.