Preheat oven to 425°F. Line a jellyroll pan or large cookie sheet with sides with aluminum foil.
Place diced pumpkin, onion, peppers, and the pecans onto a sheet pan. Drizzle with olive oil and sprinkle with salt and pepper. Toss to coat in oil. Roast for about 25 minutes or until vegetables are tender.
Meanwhile, brown the sausage in a large skillet over medium-high heat, breaking it up as it cooks. Remove it from the skillet with a slotted spoon and set aside. Add the potatoes to the skillet and cook them in the sausage drippings. Avoid stirring the potatoes so they can crisp up on the outside. Stir the potatoes once to cook through.
Return the sausage to the skillet along with all of the roasted vegetables and nuts and stir gently; remove from heat.
Melt the butter in a small skillet. Add the sage leaves and fry for one minute on each side or until crisp. Sprinkle with coarse salt; set aside. Fry the eggs over medium-high heat using the skillet and butter used to cook the sage.
Place the hash on plates and top with the sage leaves and fried eggs.
Substitute butternut squash for pumpkin, if desired. Check for tenderness after 15 to 20 minutes.