Melt butter in a large soup pot or Dutch oven over medium heat.
Add carrots, celery, shallot, and dried thyme.
Season with salt and pepper then saute until vegetables are tender, 8-10 minutes (place a lid on top to help speed up softening if vegetables are taking too long.)
Add garlic then saute for 30 more seconds.
Sprinkle in flour then saute for 1 minute.
Slowly pour in chicken broth while whisking to create a smooth sauce.
Add milk then turn heat up to medium-high and bring soup to a simmer, stirring frequently.
Turn heat down to medium then cook, stirring frequently, until soup is slightly thickened, 3-4 minutes.
Add half the mashed potatoes then whisk until smooth.
Add remaining mashed potatoes then whisk until smooth.
Remove pot from heat then stir in small handfuls of cheese one at a time, stirring until completely melted before adding more.
Stir in ham then taste soup and add salt and pepper if necessary (remember ham will add a bit of salt to each bite of soup.)
Let sit and thicken off the heat for 10 minutes then serve.
Velvety ribbons of sharp cheddar suspend within a savory broth while tender cubes of ham provide a distinct, salty bite against soft root vegetables. When constructing starch-thickened soups, the starch molecules gelatinize when heated in liquid, but adding pre-prepared mashed potatoes alters this dynamic by introducing cells that have already ruptured, granting an exceptionally smooth body without requiring a long simmer. Thoroughly whisking the chicken broth and milk prevents sedimentation while the roux cooks out its raw flour flavor over medium-high heat.
The Physics of Emulsification
Cheese incorporation requires precise thermal control because proteins tighten and expel their fat when subjected to excessive heat. Removing the pot entirely from the heat source before adding freshly shredded sharp cheddar in small handfuls ensures the dairy melts into a stable emulsion rather than breaking into an oily slick. Whisking vigorously after each addition distributes the casein proteins evenly throughout the continuous liquid phase.
Second-Day Transformations and Storage
Storing leftovers in an airtight container for up to three days changes the texture as the starches continue to retrograde and absorb free moisture. Reheating requires gentle, low-temperature application on the stovetop accompanied by a splash of milk to restore the initial viscosity without scorching the bottom of the Dutch oven.