Heat a large skillet over medium heat with a drizzle of olive oil. Add sausage and brown, crumbling it with a wooden spoon.
Add celery and butternut squash. Stirring occasionally, cook until butternut squash is golden and tender, about 10 - 15 minutes.
Add the water, thyme, vinegar, Worcestershire sauce, mustard and black-eyed peas. Let simmer for about 2 minutes and then reduce heat to low.
Heat a small skillet with several inches of water in it, until it is just barely boiling. Add a teaspoon of vinegar. Crack the eggs in into a small bowl and gently pour them into the water. Cook for about 2 minutes to keep a nice yolk. Scoop eggs out with a slotted spoon.
Divide hash between four plates and top with an egg and a generous grinding of fresh black pepper.
Notes
Andouille Sausage - can use mild pork or turkey sausage or cooked leftover meat.
Celery - can also include onion, leeks, green bell peppers.
Butternut squash - sweet potatoes or other winter squash will also work. For a summery version use zucchini.
Fresh thyme - can use dried - use ½ teaspoon dried.