Put potatoes in salted water and boil until tender, 12 to 15 minutes. Drain and smash with the milk. Mix in the bread crumbs and 1 of the eggs and season with salt and pepper, to taste. Pack into 4 (3-inch) patties.
Preheat oven to 300 degrees F.
In a nonstick pan, heat 2 tablespoons of the olive oil over high heat until the oil shimmers. Sear the potato cakes until crispy and browned, 3 to 5 minutes per side. Transfer to a sheet pan to keep warm in a low oven.
Add the spinach to the pan, and cook, season with salt and pepper, and toss until just wilted, 1 to 2 minutes. Transfer to a bowl. In the same pan, add the remaining 2 tablespoons of oil and cook the onions, garlic and thyme until soft and slightly brown, 10 minutes. Add the tomatoes and olives and cook until warmed. Transfer to a bowl and cover with foil to keep warm.
Melt the butter in the same pan, and fry the eggs, sunny-side up or as desired, about 4 minutes. Season with salt and pepper. While the eggs fry, divide the potato cakes among 4 plates. Top with the wilted spinach. Top each cake with a fried egg, and spoon warm tomato sauce over and around the egg.
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