In a large deep skillet or stockpot, combine the chicken, chicken broth and quartered onion. Bring to a boil over high heat, cover, reduce the heat and simmer for about 30 minutes, until the chicken is cooked through. Remove from the heat and allow to cool to room temperature.
Put the carrots in a steamer basket set over 2 inches of boiling water and steam, tightly covered, for 2 to 3 minutes. Add the peas and asparagus and steam for 1 to 2 minutes longer, until the asparagus are fork tender. Transfer to a colander and rinse under cold running water. Reserve 1/4 cup of the steaming liquid. In a blender, puree the cooled broth and onion until smooth.
Preheat the oven to 375 degrees F. In the skillet used for the chicken, cook the pureed broth over medium-high heat for 3 to 4 minutes, until slightly thickened. Add 1/4 cup milk, the potato flakes and potato starch and cook over medium heat, stirring, umtil smooth. Slowly add the remaining 1/2 cup milk and cook, stirring, for 3 to 5 minutes until thickened. Add the dill and season with salt and pepper. Shred the chicken into bite-sized pieces and add to the skillet.
In a saucepan, combine the reserved 1/4 cup steaming liquid, garlic and cubed onion and cook over medium heat, stirring, for 1 to 2 minutes, until the onion begins to soften. Add the spinach, cover and cook for 3 to 4 minutes, until the spinach wilts. Transfer to a colander set over a plate and set aside to drain.
Put the steamed vegetables in a 2 quart souffle dish and add the sauce, chicken and spinach mixture. Drop biscuit sized pieces of dough over the top so that they cover the chicken and vegetables and barely touch each other. Set on a baking sheet and bake for 25 to 30 minutes, until a toothpick inserted in the center of the biscuit topping comes out clean.