Lay out the trout so the skin side is down. Open the flesh like a book. With a sharp knife, carefully cut out the backbone, making sure to keep the fillets attached at the head and the tail. Season both sides of the trout with salt and pepper.
Lightly dredge both sides of the trout with flour, shaking off any excess. Heat a large saute pan over high heat. When the pan is hot, add enough olive oil to lightly coat the bottom of the pan. Heat the oil until it shimmers when swirled. Carefully add the trout, flesh-side down, starting at the tail and laying away from you towards the head. You may need to cook the trout in batches. Cook until golden brown on both sides, about 2 to 3 minutes per side. Remove to a platter and keep warm.
Once the trout is all cooked, return the pan to medium-high heat and remove all but 1 tablespoon of the oil. Add 2 tablespoons of the butter and cook just until it begins to brown and smell nutty, about 3 minutes. Add the capers, lemons, croutons, white wine, and lemon juice and cook, stirring, for 2 minutes. Add the remaining tablespoon of butter and remove from the heat. Stir in the parsley and pour over the trout.
Heat a large saute pan over high heat. Add the olive oil and heat. When the oil is hot, add the bread cubes and cook, stirring, until toasted and golden brown, about 10 minutes. Season with salt and pepper. Remove from the heat and cool.