Add the stock to a small bowl or measuring jug and sprinkle the gelatin over the top of it. Set aside until needed.
Pat the chicken breasts dry with paper towels and season generously with salt and pepper. Add the flour to a shallow bowl and dredge the chicken in it, making sure it's completely covered, before shaking off the excess.
Add a tablespoon of olive oil to a large skillet (see note 2) and set over medium-high heat. Once shimmering, add the chicken breasts. Cook on one side until golden and the chicken releases easily from the pan, about 3 minutes, before flipping and cooking on the other side. Remove from the pan and set the chicken on a paper towel-lined plate.
Reduce the heat to medium and add another tablespoon of olive oil. Add the mushrooms along with a generous pinch of salt. Cook the mushrooms until they've reduced significantly in size and are deeply golden brown, about 10 minutes.
Add the shallot and cook, stirring frequently, until it's softened, about 2 minutes. Stir in the garlic, thyme and miso paste and cook for 1 minute longer, just until fragrant and the miso is incorporated.
Pour over the chicken stock and gelatin mixture, scraping up any browned bits that have collected on the bottom of the pan. Bring to a simmer. Whisking constantly, add the butter one piece at a time, ensuring that it's fully melted before adding another piece.
Stir in the sherry vinegar and fish sauce. Add the chicken back into the pan and cook just to warm it through. Serve immediately.
Notes
I use unsalted chicken stock in all my recipe development as it allows me to control the amount of salt going into the dish.
Ideally do not use a non-stick pan for this recipe as fond collection is necessary for a proper pan sauce.