Begin by peeling the Yukon Gold potatoes and cutting them into uniform ½-inch cubes. Rinse the diced potatoes thoroughly under cold water to remove excess starch — this step helps ensure the soup maintains a smooth, creamy texture without becoming gummy. Set the potatoes aside once rinsed and drained.

In a large pot or Dutch oven set over medium heat, add the butter (or the combination of olive oil and butter). Once the butter is fully melted and starts to shimmer, add the finely chopped yellow onion. Sauté for 5 to 7 minutes, stirring occasionally, until the onion becomes soft and translucent without taking on color.

Stir in the minced garlic and cook for an additional 1 to 2 minutes, stirring constantly until the garlic becomes fragrant. Be careful not to overcook or brown the garlic, as it can turn bitter if burnt.

Sprinkle the all-purpose flour evenly over the sautéed onions and garlic. Stir constantly to coat everything and cook the mixture for 1 to 2 minutes. This forms a roux that eliminates the raw flour taste and will thicken the soup base.

Slowly pour in the chicken or vegetable broth, whisking as you go. Add it gradually to avoid clumps and to deglaze the pot — scraping the bottom to release any browned bits (fond) that add extra flavor. Continue whisking until all the broth is incorporated and the mixture is smooth. Bring the soup to a gentle simmer.

Add the cubed, rinsed potatoes to the simmering broth. Stir well, bring the mixture back up to a low simmer, then reduce the heat to medium-low. Cover the pot and let the soup cook for 15 to 20 minutes, or until the potatoes are fork-tender and easily pierced. Stir occasionally to prevent the bottom from sticking.

To achieve a creamy texture while keeping a hearty feel, blend part of the soup.

If using an immersion blender: Carefully pulse the blender in the pot several times, blending about half to two-thirds of the soup until creamy, while leaving chunks of potato intact. Be cautious of hot splatters.

If using a traditional blender: Ladle about half of the soup — both solids and liquid — into the blender. Remove the center cap from the lid and cover the opening with a folded kitchen towel to allow steam to escape safely. Blend until smooth, then return the blended soup to the pot and stir to combine. Never fill the blender more than halfway with hot liquids and always vent the lid to avoid pressure build-up.

Lower the heat to its lowest setting. Gradually pour in the whole milk or half-and-half, and the heavy cream if using, stirring gently to incorporate. Add the chopped fresh parsley, chives, and thyme (or dried thyme if using). Stir everything together and let the soup warm through gently for about 5 minutes. Avoid letting it boil at this stage, as excessive heat can cause the dairy to curdle.

Remove the pot from the heat completely before adding cheese. Begin adding the shredded sharp cheddar one handful at a time, stirring continuously after each addition until fully melted and smoothly blended. This method prevents the cheese from becoming gritty or separating.

Taste the soup and adjust the seasoning to your preference. Add salt and freshly ground black pepper gradually — remember the cheddar already contributes saltiness. For a touch of extra warmth or smoky depth, stir in a pinch of cayenne pepper or smoked paprika, if desired.

Ladle the finished soup into serving bowls while still hot and creamy. Add your favorite toppings such as crispy crumbled bacon, extra shredded cheddar, a dollop of sour cream or Greek yogurt, a sprinkle of fresh chives or parsley, or crunchy croutons for added texture. Serve immediately and enjoy the comforting richness in every bite.