Place halved Brussels sprouts in a medium sized bowl and toss gently with a small glug of extra virgin olive oil.
Warm 1 tablespoon of extra virgin olive oil in a large skillet over medium-high heat, and place the sprouts cut side down in the pan in a single layer. Do not overcrowd the sprouts. They can be touching; however, any sprouts that do not fit in the single layer, save for another batch or another use. Season with a pinch of kosher salt and cover.
Let the brussels sprouts cook for approximately 5 minutes, then turn them with tongs. They should be a deep golden brown and blackened with spots. If the sprouts are not golden, flip them back over, and move them to the hotter part of the skillet.
Once most of the sprouts are nicely golden (should only take 1 to 2 minutes longer), add the butter, shallots, garlic and lemon zest to the skillet, tossing to combine. Cook for 1 1/2 minutes, then add the white wine. The skillet will sizzle! Cook until most of the liquid has evaporated (2 to 3 minutes more). Season to taste with salt and pepper.
Transfer sprouts to a large plate and serve with lemon wedges.