Begin by melting the butter in a saucepan over low heat. Make sure that the butter doesn't brown as you prepare the next steps.
Once the butter has melted, whisk in the flour to create a roux. Cook this mixture for about 2-3 minutes, stirring constantly until it becomes bubbly and golden in color.
Gradually add the whole milk to the roux, whisking well after each addition to maintain a smooth mixture. Continue to cook the soup, stirring frequently, until it thickens, which should take about 5 minutes.
Finally, stir in the shredded cheddar cheese, mixing until it's completely melted and incorporated into the soup.