On large holes of a box grater, shred potatoes into a large bowl filled with cold water. Stir until water is cloudy, then drain. Cover potatoes with more fresh cold water. Stir to remove excess starch. Drain potatoes and spread on a clean tea towel. Roll up the tea towel with potatoes and squeeze out any excess moisture.
Mix potato shreds with onion, peppers, and jalapeño, paprika, and cayenne, if using.
In a 12-inch (30 cm) non-stick skillet, heat butter and oil over high heat. Sprinkle potato mixture into the skillet evenly. Season with salt and pepper. Cook until potatoes form a brown crust on the bottom (about 5 minutes); stir. Cook until potatoes are browned all over (about 15 minutes), stirring occasionally.
Reduce heat to low; with the back of a spoon, make 6 evenly spaced wells in the potato mixture. Crack an egg into each well. Cover and cook until whites are set, 4 to 5 minutes. Remove from heat and sprinkle with cheese.