Slice the chicken breasts in half, lengthwise and season all sides with salt and pepper. Add the tapioca starch to a large bowl and season with a pinch of salt and pepper. Set aside.
Preheat a non-stick or stainless-steel pan over medium heat. Add the olive oil and heat until shimmering. Lightly dredge both sides of each chicken breast in the tapioca starch and carefully add them to the hot oil. Cook, undisturbed, until golden brown, 4 minutes. Flip and cook until golden brown and the thickest part of the breast reaches 165F, 4 more minutes. Note, depending on size of skillet you may have to work in batches. Transfer browned chicken to a serving platter and cover with foil to keep warm.
Reduce the heat to medium-low, add the ghee and mushrooms. Cook, undisturbed, until the mushrooms start to turn golden brown, around 4 minutes. Flip and cook, stirring, until the other sides start to take on some colour, 2 minutes. Add the garlic and cook, stirring, 60 seconds.
Pour in coconut milk and use a wooden spoon to scrape off any brown bits that formed on the bottom of the pan. Bring the sauce to a simmer and reduce by half – around 4 minutes. If the sauce becomes too thick, adjust with more coconut milk, chicken broth or water. Taste and season with salt and pepper as desired. Remove from heat, add the tarragon and stir to combine.
Spoon the sauce overtop of the chicken, garnish with chives and serve immediately.