Place the mason jars on a canning rack, wire rack, or cloth, in a deep skillet. Pour water into the pan until it reaches three-quarters up the mason jars. Heat the water on medium-high and bring to a soft boil.
Heat a frying pan on medium-high. Pour in olive oil, then add the shallots and garlic . Cook until softened, about 2 minutes, then add mushrooms and sauté for 3 minutes, stirring often. Add the kale, season with salt and pepper, and continue to cooking, stirring occasionally, until tender, about 3 more minutes. Transfer the vegetables to a bowl to cool.
To the bowl of vegetables, add ⅓ cup of feta and the cream. Season with salt and pepper, and stir to combine.
Remove the mason jars from the water and cool for a couple minutes. Grease the inside of the mason jars with a light layer of butter. Spoon one-quarter of the vegetables and cream into the base of each jar, then crack an egg directly on top. Sprinkle each with 1 tablespoon of feta cheese.
Reduce the heat on the water so it is gently boiling. Carefully place the mason jars into the water and cook for 10 minutes, until the whites are set but yolks still runny. If you find the egg whites aren't cooked at the top, cover the pot for 1 to 2 minutes.
While the eggs are cooking, heat the skillet on medium. Butter both sides of each slice of bread. Fry the bread until golden and crisp on each side, about 5 minutes total. Slice into 1-inch strips.
Once the eggs are cooked, carefully remove them from the water and dry and excess moisture from the bottom. Place them on a clean plate and serve with a side of fried bread strips.