Lay scallops on a paper towel-lined plate and bring to room temperature, about 15 minutes. Season with salt and pepper and let stand 5 more minutes. While scallops rest, halve bok choy lengthwise (cut into quarters if large). Thinly slice green onions. Peel and mince garlic and ginger.
Heat a large (not nonstick) frying pan over medium-high heat until hot. Swirl in oil, then arrange scallops in a single layer in pan with ample space around each. Cook scallops, without moving, until golden and crusty underneath, about 2 minutes; then turn and cook, 1 minute more. Remove from heat and let sit in pan 1 minute to finish cooking through, then transfer to a plate.
Return pan to medium-high heat and add prosciutto in a single layer. Cook until crisp, in batches if needed, about 2 minutes, turning once. Drain on a paper towel-lined plate to cool.
Add enough oil to pan to make 1 tbsp. fat. Add green onions, garlic, and ginger and cook, stirring, until fragrant, about 30 seconds. Increase heat to high and add bok choy; cook, stirring frequently, until bok choy stems are tender-crisp, about 4 minutes. Add soy sauce and chile flakes and cook until most of soy sauce is gone, no more than 1 minute.
Spread bok choy on a large platter; top with scallops and shards of prosciutto.
*Dry-packed scallops are free of preservatives and brown better when you cook them.