Special equipment: Butcher's twine
Begin by making the brine:
In a large container, combine all the ingredients and stir until the salt and sugar dissolve. Put the turkey breasts in the brine, making sure they are completely covered. Refrigerate at least 1 hour or overnight.
To make the turkey:
Melt the butter in a large skillet over medium heat. Add the shallot, sage, thyme, chili flakes, and a pinch of salt; saute for 5 minutes. Turn the heat to medium-high. Add the pears, spreading out in a single layer. Let sit untouched for 2 minutes to brown slightly. Toss, saute 1 minute more, then remove from heat. Transfer the pear mixture to a bowl and add the walnuts, bread crumbs and chicken stock. Stir to combine and season with salt and pepper, to taste. Set aside to cool.
Remove the turkey breasts from brine and pat dry. Butterfly the turkey by slicing the breasts into even halves on the horizontal, leaving about 1/2-inch, uncut at 1 edge so the breasts can open like a book. Season all over with salt and pepper, to taste. Lay a large piece of plastic wrap on a clean cutting board. Put 3 slices of bacon parallel, about 1-inch apart on top of the plastic. Arrange 1 of the open turkey breasts, cut side up, lengthwise on top of the bacon. Scoop half of the stuffing onto 1 half of the turkey breast, fold over the other half, then roll the bacon around it tightly. Make sure the bacon strips lay flat against the turkey. Sprinkle the outside of the bacon with a little paprika. Wrap tightly in the plastic wrap twisting the ends to really seal in the breast to keep its shape. Repeat with remaining bacon, turkey, and stuffing. Transfer the breasts to a sheet tray and rest in the refrigerator for at least 1 hour.
Preheat the oven to 375 degrees F.
Remove the turkey breasts from refrigerator 1/2 hour before you plan to cook them. Discard the plastic wrap and using 3 pieces of butcher's twine, tie the twine over the center of the 3 bacon strips to secure the breasts when searing. Warm a large skillet over medium-high heat and add 1 tablespoon of the vegetable oil. Add 1 of the turkey breasts, seam side down, into the skillet and sear until the bacon is slightly crisp and the turkey is lightly browned, about 4 minutes on each side. Transfer the turkey breast to a baking sheet lined with a rack. Sear the remaining turkey breast, using 1 tablespoon of vegetable oil and add it to the baking sheet. Put the baking sheet into the oven and bake until the turkey is cooked through and the internal temperature registers 160 degrees F on an instant-read thermometer, about 60 minutes. Remove the turkey from the oven to a cutting board and let rest for 15 minutes before slicing. Cut away the butcher's twine, slice crosswise, arrange on a serving platter and serve.