Bring a large pot of water to a gentle simmer. Crack each egg into a small bowl. Create a gentle whirlpool in the simmering water and carefully slide each egg into the center. Poach for about 3 minutes for a soft yolk. Remove with a slotted spoon and drain on a paper towel.
In a small saucepan, melt the butter over medium-low heat until it turns golden brown and develops a nutty aroma, about 3-4 minutes. Be careful not to burn the butter.
Place the toasted English muffin halves on serving plates. Top each half with a handful of fresh arugula. Carefully place a poached egg on the fresh greens, then add the lobster meat. Drizzle the brown butter generously over the egg and lobster.
Sprinkle salt, freshly cracked black pepper, and the lemon zest over the top and garnish with fresh chives or parsley if desired.