Step 1: Build the Flavor Base
In a large Dutch oven or heavy-bottomed pot, melt the 4 tablespoons of unsalted butter over medium heat. Once the butter is melted and begins to foam slightly, add the diced yellow onion. Sauté, stirring occasionally, for 5-7 minutes, or until the onion is soft, translucent, and fragrant. This step isn’t just about cooking the onion; it’s about drawing out its natural sweetness to create the foundational layer of flavor for the entire soup.

Step 2: Add Aromatics and Create the Roux
Add the 4 minced garlic cloves to the pot with the onions. Cook for another minute, stirring constantly, until the garlic is fragrant. Be careful not to let it brown, as burnt garlic can taste bitter. Now, sprinkle the 1/4 cup of all-purpose flour over the cooked onions and garlic. Stir continuously for about 2 minutes to cook out the raw flour taste. This mixture of butter, aromatics, and flour is called a roux, and it is the essential thickener for our creamy soup. It should look like a thick, golden paste.

Step 3: Deglaze and Form the Soup Base
While stirring or whisking constantly to avoid lumps, slowly pour in about 1 cup of the chicken broth. Scrape the bottom of the pot to release any browned bits of onion and garlic (this is called deglazing, and it adds a ton of flavor). Once the mixture is smooth, gradually whisk in the remaining 3 cups of broth until fully combined.

Step 4: Cook the Potatoes
Add the 2 lbs of diced potatoes to the pot. Bring the mixture to a gentle boil, then immediately reduce the heat to a low simmer. Cover the pot and let it simmer for 15-20 minutes, or until the potatoes are fork-tender. You should be able to easily pierce a potato chunk with a fork with little resistance.

Step 5: Add the Ham and Broccoli
Once the potatoes are tender, stir in the 2 cups of diced ham and the 3 cups of bite-sized broccoli florets. Continue to simmer the soup for another 5-7 minutes. You want the broccoli to become bright green and tender-crisp. Be careful not to overcook it, as mushy broccoli is undesirable. This is also the perfect time for the ham to warm through and infuse its smoky, salty flavor into the broth.

Step 6: The Creamy, Cheesy Finale
Reduce the heat to the lowest possible setting. It’s very important that the soup is not boiling when you add the dairy, as high heat can cause the cream and cheese to curdle or separate. Slowly pour in the 1 cup of heavy cream and stir to combine. Now, add the 2 ½ cups of freshly shredded sharp cheddar cheese, one handful at a time, stirring gently after each addition until it is completely melted and the soup is smooth. Continue stirring until all the cheese is incorporated.

Step 7: Season and Serve
Turn off the heat. Stir in the 1/2 teaspoon of freshly ground black pepper. Now, taste the soup. Given the salt from the broth, ham, and cheese, it may not need any additional salt. If you feel it does, add a small amount at a time, stirring and tasting until it reaches your desired level. Let the soup rest for a few minutes before ladling it into bowls.