Place carrots, potatoes, and onion in bottom of slow cooker.
Mix together garlic, Italian seasoning, salt, thyme, lemon pepper, and sage and rub on outside of pork roast.
Place pork roast on top of vegetables in slow cooker.
Place halved lemon in slow cooker and pour chicken broth over everything. Cover and cook on low for 6-7 hours or on high for 3-4 hours.
Prepare broccoli as directed on package, then stir into slow cooker when it's finished cooking and serve immediately.
If desired, whisk together cornstarch and water with remaining juices and serve as a gravy on top of pork roast.
Notes
One-Pot Meal: This recipe is designed so that the pork, potatoes, and carrots cook together. However, broccoli is added at the very end (steamed separately first) so it doesn't become mushy.
Cooking Time: For the most tender "fall-apart" results, the sisters recommend cooking Low and Slow (6–7 hours) rather than on High.
Prevent Dry Pork: Pork roast can dry out if overcooked. Check the internal temperature with a meat thermometer; it is safe to eat at 145°F, though many prefer it closer to 160°F for a traditional "roast" texture.
Vegetable Sizing: Cut your potatoes into quarters and carrots into 1-inch chunks. If you cut them too small, they may turn to mush before the pork is fully cooked.
Storage Suggestions: Once cooled completely, store in an airtight container for 2 to 3 days.
Leftovers: This pork makes excellent sandwiches the next day. Shred any remaining meat and serve it on rolls with a bit of the leftover gravy or BBQ sauce.