Shred the potatoes, on the largest holes of a box grater, over a large bowl of cold water. Strain and transfer the shredded potatoes to a clean flour sack dish towel. Ring out all the water and add to a large bowl.
Whip the egg whites in a small bowl until soft peaks form. Fold in the potatoes, lemon juice and 1/2 teaspoon salt.
Coat the bottom of a large nonstick skillet with oil and heat over medium-high heat. Carefully add the potato mixture to the skillet and gently pat down. Cook until golden brown on the bottom, about 7 minutes. Top the skillet with a plate and while holding the two securely together, flip the hash brown out onto the plate. Add a bit more oil to the skillet and heat over medium-high until shimmering. Slide the hash brown back into the skillet and cook the other side until crisp and the entire potato cake is warmed through, 5 to 8 more minutes. Remove from the pan and cut into wedges.
Meanwhile, melt the butter in a large skillet over medium-high heat. Add the ham and cook, stirring, until brown and starting to crisp, about 3 minutes. Remove ham and set aside. Add the bell peppers and cook, stirring, until tender and beginning to brown, about 5 minutes. Add the ham back to the skillet. Add the green onions and stir to combine.
Top wedges of hash brown with the ham-and-pepper mixture. Dollop with sour cream and sprinkle with chives.