Place ham in a food processor; pulse until minced, about 12 times. Transfer to a large bowl; add breadcrumbs, mayonnaise, egg white, ginger, and red pepper and mix lightly but well. Form into 4 (3/4-inch) cakes; place on a wax paper-lined plate. To make ahead, refrigerate, covered, up to overnight.

Heat oil and butter in a large heavy skillet over medium-high heat until shimmering. Cook cakes until golden, about 3 minutes on each side. Drain on paper towels.

For Hollandaise sauce, process red bell pepper and lemon juice in a blender or food processor until smooth. Place butter in a small microwave-safe bowl. Cover with waxed paper and microwave on 50% power to melt, about 1 minute. Let cool slightly. Place egg yolks, red pepper mixture, and salt in a 4-cup microwave-safe bowl. Whisk until smooth. Whisk in cooled, melted butter until emulsified. Microwave on 70% power thickened, about 2 minutes, whisking briskly every 20 seconds. The mixture will resemble soft custard.

To serve, place a ham cake on each serving plate; top with poached eggs and avocado slices. Spoon Red Pepper Hollandaise Sauce on eggs and garnish with pepper and chives.