Melt butter in a large pot over medium high heat.

Add onion and garlic and cook until translucent.

Sprinkle flour over mixture and stir to form a roux. Cook several minutes to get rid of raw flour taste.

Gradually add milk while whisking until roux is fully dissolved in the liquid. Now add half and half and chicken broth while stirring.

Add potatoes, salt and pepper. Stir to combine.

Reduce heat and allow soup to simmer for about 30 minutes, stirring occasionally.

When potatoes are tender, either mash with a potato masher, use an immersion blender, or process in batches in a regular blender to puree soup.

With soup in pot, set temperature to low and add in shredded cheese a little at a time allow each to melt in.