Place the tapioca flour, 2 tablespoons grated parmesan, salt, and cracked pepper in a shallow bowl and mix to combine.
Halve the chicken breasts horizontally to make 4 fillets, then dredge each fillet in the flour mixture, shaking off excess; set aside.
Heat 1 tablespoon of the reserved sun-dried tomato oil (or olive oil) in a large skillet over medium-high heat until hot.
Fry the chicken fillets in the hot skillet until golden and cooked through, about 5–6 minutes per side depending on thickness. Transfer to a warm plate.
Add the remaining 1 tablespoon of oil to the skillet and sauté the minced garlic for about 1 minute, until fragrant.
Add the drained sun-dried tomato strips and sliced mushrooms; cook until the mushrooms are just softened, about 3–4 minutes.
Reduce heat to low and pour in the evaporated milk (or chosen milk/cream). Bring to a gentle simmer, seasoning with salt and pepper to taste.
If using milk (not cream), stir the cornstarch mixed with 2 tablespoons milk into the sauce and simmer while stirring until the sauce thickens.
Stir in the remaining 1/3 cup grated parmesan and the Italian herbs (if using) and simmer for 1 minute more until the cheese melts into the sauce.
Return the cooked chicken to the skillet, spoon the sauce over the fillets, sprinkle with shredded fresh basil, and serve.