Place the pearl potatoes in a pot of salted water. Bring to a boil and cook until fork-tender. Drain the potatoes. Place them between two sheets of parchment paper and gently press down to smash them.
Heat a cast-iron skillet over medium-high heat, lightly coated with avocado oil.
Add the smashed potatoes to the pan. Cook for 3 minutes per side until golden brown and crispy. Remove from the pan and set aside. Build the sauce. In the same skillet, add the whole garlic cloves and thyme sprigs. Cook until the garlic is golden brown. Stir in the sliced shallots and cook until softened.
Pour in the lemon juice. Lower the heat slightly and add the butter one small cube at a time, stirring constantly. Continue this process for 2–3 minutes until the sauce is emulsified and glossy. Place a lid on top of the pan to keep it warm.
Season the fish fillets on both sides with kosher salt and cracked black pepper.
Coat a grill pan with avocado oil and set it over medium-high heat.
Place the fillets in the pan, skin-side down. Cook for 3 minutes, then flip and cook for another 3 minutes until the skin is crisp and the fish is opaque.
Arrange the grilled branzino and smashed potatoes on four plates.
Generously spoon the lemon-thyme butter sauce over the fish and potatoes.
