Place the turnip in a saucepan with enough water to cover the cubes; place the saucepan over medium heat; bring to a boil; cover and reduce heat to medium-low. Simmer 5 minutes; drain water and return turnip to saucepan.
Return the saucepan to medium heat. Pour enough fresh water over the turnips to cover; add the sugar. Bring to a boil; cover, reduce heat to medium-low and simmer until turnip pieces are fork tender; drain.
Melt the butter in a separate saucepan over medium-high heat. Stir the flour into the butter to make a roux. Whisk in the milk until the mixture thickens. Add the turnips and stir gently to coat. Season with salt and pepper.