Step 1Heat the oven to 400 degrees.
Step 2In a large Dutch oven or heavy-bottom pot, bring the broth to a boil. Add the carrots and reduce the heat to a simmer, cooking until the carrots are slightly softened, about 5 minutes. Add the chicken pieces and continue to simmer until the chicken is just cooked through (it will be firm and opaque), about 10 minutes. Remove from heat and strain the chicken and carrots from the liquid (discard or save the liquid for another use). When the chicken is cool enough to handle, remove and discard the skin and bones and chop the meat roughly into 1-inch pieces. Set aside.
Step 3Heat the Dutch oven over medium heat. Add the bacon and cook until it has browned and rendered its fat, stirring frequently, 8 to 10 minutes. Drain the bacon on paper towels, leaving the bacon fat in the pot. Add the fennel, leeks and onion, and continue to cook, stirring occasionally, until the vegetables are softened and slightly translucent. Stir in the chopped chicken and carrots and cook, stirring occasionally, for 5 more minutes to marry the flavors.
Step 4While the vegetables are cooking, in a medium, heavy-bottom saucepan, melt the butter over medium heat. Whisk in the flour until thoroughly combined to form a roux, then whisk in the milk. Increase the heat and stir constantly just until the mixture comes to a simmer and thickens, 3 to 5 minutes. Remove from heat.
Step 5Pour the sauce over the chicken and vegetables and stir to incorporate. Stir in the lemon juice and zest, reserved bacon, rosemary and parsley. Season with 1 teaspoon salt and one-fourth teaspoon pepper, or to taste, and add additional lemon juice if desired. Remove from heat.
Step 6Separate the chilled dough into two equal-sized balls. On a lightly floured surface, roll out each ball into rounds approximately 13 inches in diameter. Line a 9-inch deep-dish pie plate with 1 round, and fill with the chicken mixture. Cover with second round of dough, trim the edges and crimp to seal.
Step 7Brush the top of the pie gently with the lightly beaten egg and cut a few small steam vents in the top. Place the pie on a cookie sheet and bake until golden and fragrant, about 45 minutes. Cool slightly before serving.