Preheat oven to 425° F
In the restaurant we like to cook each fish in its own 10" sauté pan, if you do not have four pans you can sear each fish off skin side down and then transfer to a baking sheet to finish in the oven.
Season each fish on both sides with salt and pepper. Preheat four sauté pans until just smoking and add olive oil. Carefully place the whole fish, skin side down, in each pan and give a little shake to make sure it is not sticking. Immediately place into oven and bake for 8 to 10 minutes or until flesh is white and firm to the touch.
While the trout is cooking in the oven place garlic and oil in a cold sauté pan and heat up until garlic starts to dance slightly, but does not take on too much color. Quickly add squash and Treviso and give a quick toss until the veg is warmed through but still has a little bite to it. Remove from the heat and wait until trout is done.
When trout is cooked, drain any excess oil from the sauté pan and with a large fish spatula carefully slide onto the plate leaving the skin side down and head and tail intact. With a dish towel wipe out any bits of fish left in the pan and add shallots and butter. When butter begins to foam and turns golden brown add balsamic vinegar.
Place veg on top of the fish and drizzle with the brown butter balsamic sauce. Top with chives and a drizzle of your favorite olive oil and enjoy.