Bring 6 quarts water to boil in a spaghetti pot and add 2 tablespoons salt.
In a large saute pan, heat virgin olive oil until just smoking. Add garlic and stir until just light brown. Add all broccoli rabe at once and cook, stirring constantly until wilted and dark green (about 3 minutes). Add 1/2 cup cold water and remove from heat. Drop spaghetti into boiling water and cook according to package instructions until just al dente (about 8 to 11 minutes). Meanwhile, cut capicola into thin julienne and add to broccoli rabe mixture. Drain pasta when cooked and toss hot pasta into pan with broccoli rabe and return to heat. Add grated cheese and stir gently to coat pasta with mixture. Pour into warm serving dish and serve immediately.