Make the salsa
In a medium bowl, whisk the olive oil and vinegar. Add the scallions, onion, roasted red pepper, gherkins, egg, capers, and mustard and whisk to combine. Just before serving, stir in the parsley and season to taste with salt and pepper.
Poach the chicken
Make a sachet by tying the parsley and 1 Tbs. of the peppercorns in a 6-inch square of cheesecloth with kitchen string.
Pour 6 quarts water into a 10- to 12-quart stockpot. Add the sachet, 2 Tbs. salt, the porcini, 2 of the bay leaves, and the cheese rind, if using. Cover and bring to a boil over high heat. Add the shallots, carrots, leeks, fennel, and celery root, then return the liquid to a boil. Reduce to a simmer and partially cover. Let the vegetables cook while you prepare the chicken, 10 to 15 minutes.
Remove the giblets from the chicken; rinse them and reserve. Put 1 tsp. salt, the garlic, the remaining 2 tsp. peppercorns, and 2 bay leaves into the cavity of the chicken. Squeeze the juice from the lemon into the cavity and add the squeezed lemon halves. Tie the bird’s legs together and tuck the wings behind the neck.
Cut a 30-inch double-layer cheesecloth square and lay it on a work surface. Put the chicken in the center, breast side up. Tie two diagonally opposite corners snugly over the bird with a simple overhand knot. Tie the other corners the same way.
Cut a length of string about a yard long and thread one end through the cheesecloth topknots. Pull the ends even and tie in a secure double knot close to the cheesecloth.
Lower the chicken into the broth, keeping the string out of the pot. Loop the free end of the string around a handle of the stockpot so it doesn’t fall into the pot. Add the reserved giblets. Bring to a boil over high heat, then lower the heat to maintain a gentle simmer.
Cook, uncovered, until the vegetables are very soft but intact and the chicken is cooked through (an instant-read thermometer inserted through the cheesecloth into the breast should register 165°F), about 50 minutes. (If you prefer firmer vegetables, check on them earlier and remove them with a slotted spoon.)
Set a large rimmed foil-lined baking sheet close to the pot. Using the string, lift the chicken out of the broth, letting the excess liquid drain into the pot. Transfer the chicken to the baking sheet. Using a slotted spoon, transfer the vegetables to a bowl and ladle a bit of the broth over them. Cover to keep warm.
Unwrap the bird, remove the string, and discard the ingredients from the cavity. Douse the chicken with hot broth and cover it with foil to keep warm.
Set a fine strainer over a large pot or bowl and strain the broth.
Carve the chicken and serve with the vegetables, moistened with a splash of the broth, and topped with the salsa.
The salsa can be made up to 5 days ahead. Cover and refrigerate, then let sit at room temperature for 30 minutes before serving.
The leftover broth can be refrigerated for up to 5 days or frozen for up to 3 months.