Place 4 Mini Round Cocottes on a small baking sheet and add 1 tbsp (15 ml) of the oil to each one. Cover the cocottes and transfer, on the baking sheet, to the lower rack of the oven. Preheat the oven to 425°F (220°C).
In a strainer set over a large bowl, toss the grated potato and sweet potato with a generous amount of salt and a few cracks of pepper. Let drain.
As soon as the oven is preheated, carefully divide the potato mixture among the cocottes, packing it down into the hot oil with the back of a spoon. Place a small knob of butter (about 1 tsp/5 ml) into each cocotte. Bake, uncovered, for 15 minutes. Stir and bake for another 15 minutes or until tender and crispy around the edges.
Meanwhile, in a Iron Handle Skillet or Saucier, melt the remaining butter over medium heat. Add the mushrooms and sauté until tender. Season with salt and pepper. Add the baby spinach (or other green of your choice) and chives. Season lightly with salt and pepper. Sauté until the greens are slightly wilted. Remove from the heat and stir in the Parmesan. Adjust the seasoning.
Remove the cocottes from the oven. Add a slice of ham to each cocotte, cutting the slices to fit. Top with the mushroom mixture. Make a well in the centre. Carefully crack an egg into each well. Season with salt and pepper.
Bake for 12 to 15 minutes, or until the egg whites are set but the yolks are still runny. Sprinkle with more chives and Parmesan, to taste. Serve immediately with grilled or toasted bread for dipping into the runny egg yolk.