Preheat the oven to 375 degrees .
Heat a medium cast-iron or other ovenproof skillet over medium. Add the oil, two turns of the pan. Add the shallots and garlic; cook, stirring often, until aromatic, 2 to 3 minutes. Add the stock and bring to a bubble. Add the spinach and cook, stirring often, until heated through, about 1 minute. Add the baby kale and stir until wilted, 1 to 2 minutes; season with salt, pepper and nutmeg. Add the Boursin and stir until melted, 2 to 3 minutes. Add the cream and simmer until the mixture thickens a bit, 2 to 3 minutes. Crack the eggs, 1 at a time, into a ramekin or small bowl to check for shells, then carefully add the eggs to the skillet, spacing them apart evenly and nestling them in the creamed spinach. Top with the cheeses and bake until the egg whites are set but the yolks are still runny, about 10 minutes or until set to your liking. Serve from the skillet, scooping into shallow bowls. Serve with bread for dipping.