Rinse trout inside and out with cold running water; pat dry with paper towels
Soak trout in milk 10 minutes, turning the trout over halfway through this time.
On waxed paper, combine flour and salt. Remove trout from milk and coat evenly with flour mixture, shaking off excess.
In a medium skillet, heat 2 tablespoons oil over medium heat until very hot. Add 2 trout and cook until just opaque throughout, about 5 minutes per side.
Transfer to platter and keep warm. Repeat with remaining 2 tablespoons oil and remaining fish.
Pour off all but 1 tablespoon fat remaining in skillet.
Return skillet to heat, and add lemon juice, then add butter. Cook until the butter and lemon juice are slightly foamy, about 2-3 minutes.
Stir in parsley, pour the butter sauce over fish. Sprinkle with sea salt if desired.