Combine ham hock, onions, carrot, water, chicken base, sage, thyme, and rosemary in a large slow cooker. Cover and cook on Low, 8 hours to overnight.
Stir potatoes and tomatoes into the soup and cook on Low, 3 to 4 hours more.
Remove ham hock, shredding with 2 forks, leaving behind any meat. Add cubed ham.
Meanwhile, melt butter in a saute pan over medium heat. Whisk in flour until it is thick, aromatic, and slightly browned, 15 to 20 minutes. Add half-and-half until it is gravy consistency. Add mixture to the slow cooker and mix well. Serve with Cheddar cheese.
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