Make the sauce first. Heat the olive oil in a skillet over medium-high heat. Add the shallots and the shrimp and cook until the shrimp turn pink. Lower the heat to medium.
Add the wine, lemon juice, and stock, and bring to a boil. Reduce the liquid to about one-fourth the original volume, then lower the heat to as low as it will go.
Cut the stick of butter into pats and whisk them in to make a creamy-looking sauce. Add the crabmeat, and agitate the pan until the crabmeat is heated through. Cover the pan and turn off the heat.
Mix the salt and pepper into the flour. Dust the trout fillets liberally in the seasoned flour.
Heat the 1/2 cup of butter over medium-high heat until it shimmers. Saute the trout, two fillets at a time, until golden brown--about three minutes per side.
Place a trout fillet on each serving plate and top with the sauce.
