Place the bacon in the braiser and cook over medium heat until browned, about 8 minutes. Transfer with a slotted spoon to drain on paper towels. Reserve the fat in the pan.
Pat the chicken dry and season with salt and pepper. Working in batches, add the pieces to the pot skin side down and cook undisturbed over medium heat until browned, about 3 minutes. When sufficiently browned, the chicken will release easily when lifted with tongs. Flip and brown the other side. Transfer to a plate.
Add the onions and parsnips and cook until they begin to brown, stirring occasionally, about 8 minutes. Deglaze the pot with the sparkling wine, stirring to loosen the fond from the bottom of the pot. Add the bay leaf and thyme and bring to a boil.
Return the chicken thighs and legs to the braiser. Cover, reduce the heat to low, and simmer for 1 hour. Refrigerate the chicken breasts until needed.
Return the chicken breast pieces and bacon to the braiser. Cover and simmer until tender, about 40 minutes. Transfer the chicken to a warmed serving platter and cover loosely with foil to keep warm while you prepare the Haricot Verts and sauce.
Discard the thyme sprigs and bay leaf. Bring the sauce to a rapid boil and let reduce until slightly thickened, 5 to 8 minutes. Remove the pot from the heat and swirl in the crème fraiche and 2 ¼ teaspoons of the dill. Season with salt and pepper.
Spoon the vegetables and sauce over chicken and garnish with the remaining dill.
To prepare the Charred Haricots Verts:
Heat a large cast iron skillet over medium-high heat, then brush with the canola oil. The oil should shimmer but reduce the heat if it smokes. Arrange the shallots cut side down in the skillet and let sear until deeply browned on both sides, 4-5 minutes. Transfer to a plate.
Working in batches, add the beans to the skillet and cook, tossing occasionally, until crisp-tender and lightly charred in spots, about 8 minutes. Transfer to a large bowl and cover with foil to keep warm while cooking the next batch.
Add the butter to the beans ½ tablespoon at a time, alternating with half of the lemon juice. Toss to incorporate. Add the shallots and half each of the lemon zest, almonds, mint and parsley. Toss to mix and season with the salt and pepper.
Transfer the beans to a serving platter. Sprinkle with the remaining lemon zest, almonds, mint and parsley. Serve warm.