In a large heavy skillet heat 2 tablespoons of the butter over medium-low heat until melted. Add the garlic and stir just until fragrant, about 30 seconds. Add the fiddleheads and squash, and broth. Increase the heat to medium-high and bring to a simmer; cover and simmer for 1 minute. Add the peas, ramp tops and bottoms, and mint. Cover and simmer for 1 minute, then stir in the remaining 2 tablespoons butter, until the butter is just melted. Season the ragout with salt and pepper.
Meanwhile, heat another large heavy skillet over medium-high heat. When it is hot enough that a few drops of water sizzle and evaporate very quickly (like less than 5 seconds), add the vegetable oil and tilt the pan to even coat the surface. Season the skin side of the fish (the rounded side) and place the fillets skin-side down in the pan. Season the other side. Cook until lightly golden on the skin side, about 4 minutes. Carefully flip them and cook on the other side until cooked through but still translucent in the center, about 3 minutes more.
To make a fancy presentation, place a pile of the ragout at about 10 o'clock on each of 4 plates. Place the fish on a diagonal (with the bottom side facing about 5 o'clock), with one corner slightly resting on top of the ragout. Sprinkle with parsley and serve.
To clean fiddleheads:
Soak the fiddleheads in a sink half full of cold water, lifting them out to leave the water behind and changing the water several times to remove any grit or casing particles. Drain thoroughly.