In a medium skillet over medium heat, melt 1 tablespoon of butter and sauté the diced onion, celery, and garlic for 5 to 7 minutes until soft and translucent.

Transfer the sautéed vegetables into a 6-quart slow cooker stoneware liner.

Add the diced Russet potatoes, diced cooked ham, freshly ground black pepper, and low-sodium chicken broth to the slow cooker, stirring to combine.

Cover and cook on LOW for 6 hours (or HIGH for 3 to 4 hours) until the potatoes are completely fork-tender.

Using a hand potato masher, mash roughly half of the potatoes directly inside the slow cooker to thicken the broth while leaving whole potato chunks intact.

In a small saucepan over medium heat, melt the remaining 2 tablespoons of butter, whisk in the flour, and cook for 1 minute until bubbling.

Gradually whisk the evaporated milk into the roux saucepan and cook for 2 minutes until hot and slightly thickened.

Pour the warm milk roux mixture into the slow cooker and stir thoroughly to combine.

Cover and cook on HIGH for 20 to 30 minutes until the soup is hot and bubbling around the edges.

Turn off the heat, stir in the freshly grated sharp cheddar cheese until melted and smooth, taste for seasoning, and add salt if needed.

Ladle into warm bowls and garnish with chopped fresh chives before serving.