Pat chicken thighs dry with paper towels then lightly season all over with salt. In a large skillet or Dutch oven, melt 1 tablespoon butter over medium-high heat. Working in batches, add chicken and cook until browned on both sides, 3 to 4 minutes per side.

Return all chicken to skillet or Dutch oven and add chicken stock, bring to a simmer, and scrape the bottom of pan with a wooden spoon to loosen any browned bits. Lower heat to medium-low and simmer until chicken registers 175°F (80℃), 5 to 7 minutes. Use tongs to transfer chicken to a cutting board to cool.

Strain the cooking liquid through a fine-mesh strainer into a heatproof measuring cup or medium bowl. Stir in milk; set aside.

In the now-empty pan, melt remaining 2 tablespoons butter over medium heat. Add onion and celery and cook, stirring occasionally, until softened and translucent, 5 to 7 minutes. Add garlic, black pepper, and salt and cook until aromatic, about 1 minute. Sprinkle flour over the vegetables and stir to coat. Cook until toasted and no dry flour remains, 1 to 2 minutes.

Gradually pour in the reserved stock and milk mixture, whisking constantly. Bring to a boil, then reduce heat to maintain a simmer and cook, stirring often, until gravy thickens enough to coat the back of a spoon, about 5 minutes.

Using an immersion blender (or carefully transferring to a standard blender), purée the sauce until smooth. Taste and adjust seasoning with salt and pepper as needed. If using a blender, return sauce to pot.

Using 2 forks, shred chicken finely, then return to pot with sauce. Simmer until mixture resembles a thick gravy and a line drawn by a spoon holds its shape for a few seconds, 7 to 10 minutes. Remove from heat, stir in reserved banana pepper brine, and let cool for 15 minutes. (It will thicken further as it cools.)

For serving sandwiches: For each sandwich, pile about ½ cup shredded chicken mixture onto a bottom bun. Add pickled banana peppers. Top with the remaining bun and serve warm.