In a heavy-bottomed pot, melt the butter over medium heat. Pat pheasant pieces dry with paper towels and season with salt. Brown the pheasant on both sides in the butter and remove it from the pot.
Add the chopped celery, onion, diced carrots, and a pinch of salt to the remaining butter. Sweat the vegetables until the onion becomes translucent, about 5-7 minutes. Then, add garlic and stir for 30 seconds until fragrant. Add the pheasant pieces back to the pot, along with 6 cups of water. Bring it to a boil and stir in the chicken base to dissolve. Lower heat to a simmer and cook covered for 1 to 1.5 hours or until the pheasant becomes tender.
Meanwhile, make the knoephla dumplings. In a mixing bowl, combine the flour, dill, parsley, salt, and pepper. Then, make a well in the middle of the flour and add one egg. Use a fork to lightly beat the egg, then pour milk into the well. Incorporate the dry ingredients into the wet ingredients, then lightly knead until you get a ball. Do not overwork the dough.
Place the dough onto a lightly floured surface and divide into four equal pieces. Working with one piece at a time, roll it into a thin log about ¾ inch in diameter. Cut the log into small, bite-size pieces. (In hindsight, I would’ve preferred smaller pieces than what I made due to their density.) Place the raw dumplings, not touching, on a cookie sheet lined with wax or parchment paper. Keep dumplings covered with a towel to keep them from drying out while you work with the remaining dough.
When tender, remove the pheasant from the soup and set it aside. Bring the soup to a boil and then drop in dumplings one by one, stirring occasionally to prevent clumping. Bring soup down to a simmer, cover, and cook for 10 minutes. Meanwhile, peel the potatoes and cut them into 1-inch cubes. Shred the pheasant into bite-size pieces and discard the bones.
Add potatoes and shredded pheasant back to the pot. Add milk and return to simmer. Cook until potatoes are tender, about 20 minutes, stirring occasionally. Do not allow the milk to boil. Season to taste and enjoy.