Season the salmon pieces with salt and pepper. Dredge in flour, shake off excess and set aside. Reserve the flour for making roux.
Preheat a large skillet or a saute pan over medium high heat. Melt 4 tablespoons of butter and 5 tablespoons of olive oil. When butter and oil start to sizzle, add the salmon pieces, skin side up, and saute for 2 minutes. Flip the salmon pieces and continue sauteing for another minute or so, until the salmon is cooked through (internal temperature is 145F). Take the pan off the heat, remove the salmon and set aside.
Return the pan to the heat and add the remaining 2 tablespoons of butter. When the butter and oil begin to sizzle, add two heaping tablespoons of reserved flour and stir well. Cook for about a minute, until the flour turns deep golden brown. Carefully pour the lemon juice and the chicken stock into the pan. Add the capers and bring to boil, scraping up brown bits from the pan for extra flavor. Taste and adjust the seasoning, if needed.
Continue simmering the mixture over low heat for about 3-4 minutes, until it thickens to your liking. Add the salmon pieces, skin side down, pour some sauce over the fish and continue simmering for another minute. Take the pan off heat, remove the salmon and set aside. Whisk the sauce and pour over the salmon pieces.
Serve immediately with pasta or steamed vegetables. Garnish the salmon with 2-3 tablespoons of sauce, lemon slices and fresh dill.