Preheat the broiler to high. Mix 4 tablespoons butter with the mustard and anchovy paste in a small bowl until smooth.
Spread the anchovy butter on the cut sides of the bread; place buttered-side up on a baking sheet and broil until toasted, 1 to 2 minutes. Set aside.
Melt the remaining 2 tablespoons butter in a large skillet over medium-high heat. Crack in the eggs; season with salt and pepper and let cook until the yolks just begin to firm up and the edges are slightly crisp, about 3 minutes.
Assemble the sandwiches: Place 2 bread halves on each of 4 plates; top with a slice of pate, an egg, some cheese, greens and chives.