Heat oil in large skillet over medium-high heat. Add pork cubes and brown on all sides, turning as needed, about 5 minutes. Transfer pork to plate, season with salt and pepper.
Add onions, carrots and apples to skillet and cook, stirring often, until onions begin to brown, about 5 to 6 minutes. Transfer to a 3 1/2-quart slow cooker and top with pork cubes.
Add apple cider, thyme, allspice and sage to skillet. Bring to a simmer, scraping up brown bits on bottom of skillet; add to the slow cooker.
Cover and slow-cook until pork is tender, 6 to 7 hours on low.
Using a slotted spoon, transfer meat to a serving bowl, cover to keep warm.
Skim off fat from surface of cooking liquid. In food processor or blender, puree cooking liquid and solids until smooth.
Pour sauce over meat, stir gently and serve immediately.