In a large, deep skillet, melt 2 tablespoons of the butter in the oil. Add the parsnips and cook over moderately high heat, stirring occasionally, until lightly browned, about 6 minutes. Add the shallots and cook, stirring, until softened, about 2 minutes. Add the stock and thyme and bring to a boil. Season with salt and pepper, cover and simmer over low heat until the parsnips are tender, about 8 minutes.
Meanwhile, fill a large, deep pot with 2 inches of water and bring to a boil. Add the spinach in large handfuls and blanch, stirring, just until wilted, about 10 seconds. Drain and cool under running water. Squeeze the spinach dry and coarsely chop it. Stir the spinach into the parsnips.
In a medium saucepan, melt the remaining 2 tablespoons of butter and cook over moderately high heat until lightly browned, about 4 minutes. Whisk in the flour and cook, whisking, for 1 minute. Whisk in the half-and-half and nutmeg, season with salt and pepper and bring the sauce to a boil, whisking until thickened, about 2 minutes. Stir the sauce into the spinach and parsnips and bring to a simmer. Transfer to a bowl and serve.