Preheat the oven to 475°F and place the rack in the middle position.
Drizzle 1 tablespoon of olive oil onto a baking sheet. Use your fingers or a pastry brush to spread it evenly across the sheet.
Add the cut potatoes to a large bowl and add the remaining tablespoon of olive oil, ½ teaspoon Kosher salt, and ¼ teaspoon ground black pepper. Toss to coat potatoes well.
Spread the potatoes onto the prepared baking sheet in an even layer. Place the baking sheet into the preheated oven and bake 20 minutes. After 20 minutes, turn the potatoes and add the raw sausages atop and in between the potatoes. Place back into the oven for 12 minutes.
While the potatoes and sausages are cooking, use 2 tablespoons of the unsalted butter to butter each slice of bread. Make sure to spread the butter over the entire surface of the bread. Use your cookie or biscuit cutter and position it directly in the middle of each slice, using a gentle touch to pop out the centers. Set all aside.
Remove the baking sheet from the oven. Give the potatoes and the sausages a good flip and push all to one half the baking sheet. Place each slice of bread and its middle, buttered side down, onto the baking sheet and return to oven for about 4 minutes.
Remove the baking sheet from the oven again. Flip the toasts over and add the remaining tablespoon of butter to the baking sheet to melt. Use a pastry brush to push it around if you need to.
Crack an egg and shimmy it into a bread hole. Repeat with remaining eggs and slices of bread. Place the baking sheet back into the oven for the eggs to set, about 4 to 6 minutes. Whites will be slightly jiggly but set, yolks creamy and slightly runny. Oven times will vary.
To assemble, ensure each plate gets ample crispy home fries, sausages, and for each egg-in-a-hole, its popped out toasty middle for dipping into the yolks. Serve hot with fresh coffee and plenty of orange or grapefruit juice.