Spray 4 (8-ounce) ramekins with the olive oil spray and set aside.
Put the chicken breasts in a medium saucepan, add the broth (it should just cover the chicken), and bring to a boil. Cover, turn the heat very low, and cook the chicken very gently until just firm to the touch, about 15 minutes. Remove chicken from the broth reserving the broth. When the cool enough to handle, shred or dice the chicken into large bite-size pieces, discarding the bones. (If some of the pieces are still a bit pink, don't fret, it finishes cooking in the pie.).
Preheat oven to 400 degrees F.
Heat the oil in a heavy non-stick skillet over medium-high heat. Add the carrots, mushrooms, celery, turnips, season with salt, to taste, and cook until the vegetables are light brown, about 8 minutes. Add the onions and cook about 1 minute more. Stir the flour into the vegetables and cook for 1 minute. Pour in the reserved chicken broth and whisk until it comes to a boil. Reduce the heat slightly and simmer, uncovered, until thick, about 5 to 6 minutes. Stir in the chicken parsley, dill, lemon zest, and pepper and remove from the heat.
Layer 3 sheets of phyllo dough on the workspace, spraying with olive oil spray in between each layer. On the last layer, sprinkle with more salt and pepper, and the paprika. Cut sheets in half, crosswise, and stack to make 6 layers. Cut 4 rounds out of the phyllo with scissors or a paring knife that fit snugly into the opening of each ramekin. Divide chicken filling evenly among the ramekins and place the phyllo rounds on top. Bake until bubbly, chicken is cooked through and crust is crispy and golden brown, about 10 to 12 minutes.