For the chicken:
Rinse chicken and pat dry. Salt and pepper both sides of the chicken.
Preheat a 5-qt. saucier over medium heat until the rim of the pan is hot to the touch. Add the oil and allow to heat for another minute. Add the chicken, about half at a time; cook, turning as needed until brown on all sides. As the chicken is browned, remove it from the pan and set aside.
Pour off all but 1 Tbsp. of the oil and fat in the pan. Reduce heat to medium. Add the onions and mushrooms to the pan and cook, stirring often until lightly browned, about 3 - 5 minutes.
Add carrots and sage and return the chicken to the pan on top of the vegetables. Pour in broth and wine. Bring everything to a boil, then reduce heat to low; cover and simmer for 20 minutes.
For the herb dumplings:
In a large bowl, combine the flour, baking powder, salt and dried dill weed.
Using a pastry blender or 2 knives, cut 5 Tbsp. of firm butter into the flour mixture until it resembles a coarse meal. Mix in the chopped herbs gently. Make a well in the center of the mixture and pour in the evaporated milk. Incorporate the milk and flour with a fork until the dough forms a ball.
Final assembly and cooking:
Skim and discard the fat from the chicken preparation. Season to taste with salt and pepper. Drop dumpling dough over chicken in 8 equal portions over the surface of the chicken, spacing slightly apart. Cover and continue to cook until dumplings are firm to-the-touch and a wooden pick inserted in the center comes out clean, about 20 - 25 minutes.