The mirepoix (carrot,celery,onion) cleaned and diced small. Now lets "cream" or puree the garlic. Simply mince garlic very fine, when minced add some salt to help break down the fibers in the garlic. Set aside.
In a heated sauce pot drizzle olive oil, when oil reaches a point were it is just smoking place mirepoix(diced carrot,celery,onion) in pot. Sweat the mirepoix until the onions are a little translucent. place the creamed garlic and combine thouroughly. Wash potatoes and dice to medium. Carefully cut potato in half. Lay potato on the flat side, this helps secure the product to prevent from slipping and possibly cutting yourself. From here slice the potato length wise. Turn the potato slice in the opposite direction to produce the dice. Place potato into sauce pot. Reason we put potatoes in after the mirepoix is developing "layers" of flavor. Now for our leftover pot roast. Simply rough cut and dice beef and place in sauce pot.
Now here's the hard part. Waiting. Saute the mirepoix, garlic and beef for a minutes to develop a "fond" (it's the residue from the sauteing products. When foods make contact with a hot surface it's called the Miarde reaction, leaving residuals of salts and sugars protiens that extracted from the products natural content, in other words it's the brown stuff on the bottom of your pan, that's were FLAVOR lives) Let the soup develop its flavor on medium heat, at this point add your liquid. Do not cover the pot, you want to have the liquid reduce slightly. Let simmer on medium het for about 45 minutes to an hour.