Bring a large pot of water to a boil. While waiting, pit and roughly chop the olives, thinly slice the garlic, and cube the cold butter. Measure out the olive brine and vermouth so everything is ready.

Salt the boiling water aggressively (about 2 tablespoons). Add spaghetti and cook until just shy of al dente, about 2 minutes less than package directions. Before draining, scoop out 1.5 cups of starchy pasta water and set aside. Drain the pasta.

Heat olive oil in a large skillet over medium heat. Add garlic and cook until it just starts to turn golden at the edges, about 2 minutes. Do not let it brown. Add the chopped olives and cook for 1 minute until fragrant.

Pour in the vermouth and olive brine. Increase heat to medium-high and simmer until the liquid reduces by half and smells slightly sweet, about 4 minutes. You want about 0.25 cup of liquid remaining.

Reduce heat to low. Add the hot drained pasta and toss to coat. Add the cold butter cubes one at a time, tossing constantly until each melts before adding the next. This creates a silky, glossy sauce. If the pan looks dry, splash in reserved pasta water 2 tablespoons at a time until the pasta is loose and creamy.

Remove from heat. Add the parmesan, lemon zest, and black pepper. Toss vigorously for 30 seconds until the cheese melts into the sauce. Taste and adjust - add more brine for saltiness or pasta water to loosen. Serve immediately in warmed bowls with extra parmesan and cracked pepper.