Preheat oven to 375°. Dice enough cucumber to make 3/4 cup. Cut remaining cucumber in half lengthwise. Scrape out any coarse seeds with a small spoon and discard. Thinly slice cucumber crosswise.
Set salmon skin side down on a work surface. Cut fish down to the skin, but not through it, to divide into 4 portions.
Line a 9- by 13-in. baking dish with foil. Put salmon skin side down in dish. Melt 1 tbsp. butter and mix with lemon juice, 1/4 tsp. salt, and the pepper; spoon over fish and into crevices.
Bake salmon until barely opaque but still moist in center of thickest part, 10 to 13 minutes. Remove from oven; keep warm.
While salmon bakes, combine shallots, chopped tarragon, and the vinegar in a 10-in. frying pan. Boil over high heat, stirring often, until vinegar evaporates but shallots still look wet, 4 to 5 minutes. Add 1 tbsp. butter, remove from heat, and stir until it melts. Stir in diced cucumber; set aside.
Transfer cooked fish to a warm platter or plates, using a wide spatula to lift it from skin. Return cucumber mixture to high heat and cook, stirring often, until slightly translucent and hot, 1 to 2 minutes. Add remaining 3 tbsp. butter and 1/4 tsp. salt. Stir just until butter melts and sauce looks creamy. Remove from heat and stir in sliced cucumber and parsley.
Spoon cucumber sauce over salmon and scatter tarragon leaves on top. Serve immediately so colors stay bright, and season to taste with more salt if you like.
*Ask your fishmonger to remove bones, or do it yourself: Run a finger over flesh to locate bones, then pull them out with fish tweezers (see how at sunset.com/salmonvideo).