Step #1 In a skillet cook 3 c of the onions in 1/2 c water over moderate heat for 15 to 20 mins, or until they are very soft.
Step #2 Add the vinegar & cook the mixture for 2 mins, or until the liquid is evaporated.
Step #3 Add the broth & let simmer, stirring for 30 mins.
Step #4 Strain the liquid through a sieve into a bowl, reserving it, & in a food processor/blender purée the onions with 1/4 c of the reserved liquid.
Step #5 In a saucepan, boil remaining reserved liquid until reduced to about 1 c.
Step #6 Stir the onion purée into the reduced liquid, & keep the sauce warm, covered, stir in the the cream, over very low heat.
Step #7 Toss the remaining 2 c onions with the flour, shake off the excess, reserving it.
Step #8 In a deep fryer heat the peanut oil to 375 degrees F, a few at a time, & fry the onions, for 1 to 1 1/2 mins.
Step #9 Transfer the onions with a slotted spoon to paper towels to drain & chop them into 1/4-inch pieces.
Step #10 In an ovenproof skillet heat the olive oil over moderately high heat until hot but not smoking & cook the salmon, skin side up and.
Step #11 seasoned with salt & pepper, for 1 1/2 mins.
Step #12 Turn the fillets carefully with a spatula & cook them for 2 mins more.
Step #13 Dip the tops of the fillets carefully into the reserved flour, coat them with the egg, & top them with the fried onions.
Step #14 Return the fillets to the skillet, covered loosely with foil wrap, spoon any remaining fried onions on top of the fillets, & bake the fillets, in the middle of a preheated 350-degree F.
Step #15 oven for 4 to.
Step #16 5 mins, or until just cooked through.
Step #17 Divide the sauce among 4 heated plates, transfer the salmon with a spatula to the plates, & spoon the caviar decoratively onto it.