Lightly season both sides of each chicken breast with salt and pepper
Preheat a 12” stainless-steel or cast-iron skillet over high heat. Add avocado oil and heat until shimmering. Cook the chicken, undisturbed, until golden brown, around 4 minutes. Flip and cook an additional 3 to 4 minutes or until golden brown on the second side. Transfer the chicken to a serving platter and set aside. You can tent it with foil to keep it warm.
Return the pan to the stovetop and lower the heat to medium. Add the mushrooms and ghee and season with a pinch of salt. Cook the mushrooms, stirring and scraping any brown bits on the bottom of the pan, until softened, 4 to 5 minutes.
Once the mushrooms have browned and their moisture has evaporated, add the onions and cook, stirring, until softened, 3 to 4 minutes. Add the garlic and cook, stirring, 1 minute.
Add coconut milk and mustard, bring to a simmer and cook, stirring regularly, until reduced by half in volume. Add the chicken stock, bring to a simmer and cook, stirring regularly, until reduced by half in volume.
Remove from heat, add dill and lemon juice and season with black pepper to taste. Stir to combine, taste the sauce for seasoning and adjust with salt and pepper as desired.
Spoon the hot stroganoff sauce overtop of the chicken and serve immediately.