In large stock pot, combine chicken, carrots, celery, onion, garlic, salt & cover with water. Bring to a boil, reduce heat, cover & simmer until chicken is fully cooked. I always let it simmer for awhile to create a good stock.
While chicken cooks, combine eggs & pinch of salt in mixing bowl & beat eggs slightly- I use a fork, because, well. I don’t have a fancy mixer. Gradually add flour & milk to eggs & continue mixing until the flour is incorporated & dough is no longer sticky to the touch. Slightly more or less flour may be necessary, depending on the size of your eggs. This is kind of a guessing game.
When dough is ready, form into a ball & turn out onto a well floured surface. Knead by hand a few times, adding more flour if it gets sticky again. Pat into a round & then roll out into a large rectangle about 12x15 or so & 1/4 inch thick. Be sure to flip dough often, keeping it well floured so it doesn't stick.
Starting at a short end, use a pizza cutter to cut the dough into 1/4 inch wide (or smaller) strips. When all strips are cut, toss them with enough flour to keep them from sticking together. Spread noodles out in a single layer on your rolling surface & leave to dry a bit while you are finishing the chicken.
When chicken is cooked, remove to a cutting board, strain broth and return to pot, discard carrots etc. & use two forks to shred chicken into bite-sized lengths.
Add the chicken broth to the stock pot & bring broth & water to a boil.
Add noodles & whatever flour is with them to the boiling pot.
Add the shredded chicken & stir gently. Reduce heat to low to medium-low and simmer, uncovered, for about 10 minutes.
Simmer all for another 15-20 minutes, stirring occasionally, or until noodles are tender. Timing really depends on how thick your noodles are cut. We like them a little on the thick side.
Serve over mashed potatoes.