Put 1/3 cup panko bread crumbs in a small saucepan over medium heat. Toast, stirring frequently, until the crumbs are golden and fragrant, about 3–4 minutes.
Turn off the heat and immediately stir in 1 minced or pressed garlic clove, 1/4 teaspoon dried basil, and 1 tablespoon brown butter. Transfer the toasted breadcrumb mixture to a plate and set aside. Reserve the remaining brown butter for finishing.
Season both sides of the 4 fresh tilapia filets with 1/2 teaspoon salt and 1/2 teaspoon pepper.
Heat a large skillet over medium-high heat. Add 1 1/2 tablespoons coconut oil and let it melt and warm until shimmering.
Add the seasoned tilapia to the skillet (do not overcrowd) and cook 2–3 minutes per side, until the fillets are slightly golden and opaque through the center. Remove the fillets from the skillet and keep off the heat.
Right before serving, spoon the toasted basil butter breadcrumbs over the tilapia and drizzle with the reserved brown butter. Serve immediately.