Put 1 Tb butter and 1 Tb olive oil in a saute/frying pan and get hot enough for butter to bubble. Add salmon skin side down. Sprinkle salmon with kosher salt, pepper and paprika. Cook for 4-5 minutes until you see the color move up the salmon about half way through the fillet. Flip fillet over and cook for another 2-3 minutes until medium/medium rare in temperature. Set salmon aside.
While salmon is cooking, prepare shallots and capers. Put the other 1 Tb butter and 1 Tb olive oil in a small pan and saute shallots until soft and caramelized, about 7 minutes. Add capers and lemon juice and stir to combine. Pour sauce over salmon when ready to serve. Garnish with lemon slices.
Toss the green beans into a pot of water at a rolling boil. Boil for 4-5 minutes until beans are bright green and slightly softened. Drain beans, toss with some olive oil and top with cheese.