Place the chicken thighs on a baking sheet and pat them dry with a clean paper towel. Season both sides generously with kosher salt. Chill 1 hour, or up to overnight.
Preheat the oven to 350°.
In a large ovenproof baking dish, place the chicken thighs in an even layer with minimal overlap. Add the shallot, garlic, celery stalk, and hot sauce, making sure to fully coat the vegetables in the sauce. Wrap the dish tightly with aluminum foil, then bake until the chicken is tender, 50 minutes.
Retrieve the dish from the oven and let sit, covered, for 10 minutes. Remove the foil and insert a paring knife into the thickest part of one of the thighs to check for doneness. (The chicken should gently fall away from the tip of the knife with no resistance.)
Using tongs, transfer the chicken thighs to a plate or a baking sheet (do not discard the cooking liquid). Set aside.
In a saucepan over medium-high heat, strain the cooking liquid and discard the solids. Bring the liquid to a boil, then immediately reduce to a simmer. Cook, stirring occasionally and skimming off any fat from the surface, until the sauce is reduced by half, about 25 minutes.
Transfer the sauce to a bowl set over an ice bath, and let cool completely, about 30 minutes.
8. In a stand mixer fitted with the paddle attachment, add the cooked chicken thighs. Mix on low speed until the thighs are gently shredded, but not completely broken down, about 2 minutes. (Alternatively, you can shred the chicken by hand.) Stop the mixer, add the diced celery, reduced sauce, and 1 ½ ounces schmaltz. Mix on low speed until the schmaltz is incorporated, about 1 minute. Rillettes can be prepared up to 1 day in advance. Remove and let come to room temperature before serving.
Make the schmaltz toast: Cut the fresh baguette lengthwise into thirds, then halve each piece crosswise. Brush the cut sides with melted schmaltz.
Warm a cast iron pan over medium heat. Working in batches of three at a time, place the pieces schmaltz side down in the cast iron pan; cook, gently pressing the pieces down as needed, until crisp and golden, 1-2 minutes. Remove and sprinkle each with a pinch of flaky sea salt. Slice the baguette pieces into 1-inch-wide toasts, and serve warm with the rillettes.