Increase heat to medium high. Add the squash to the pot with the bacon fat and cook until lightly browned, 4 to 6 minutes (resist the urge to stir it too often or it won't brown). Stir in the apple, sage, salt, and pepper and cook for about 4 minutes (you'll see more browning occur on the bottom of the pot than on the vegetables). Add the broth, scraping up any browned bits in the pot with a wooden spoon. Bring to a boil over high heat, reduce the heat to maintain a simmer, and cook until the squash and apples are very soft, 6 to 8 minutes. Remove from the heat and let cool somewhat.