In a 1-quart saucepan combine the sweet potatoes, water, sugars, 2 tablespoons of the butter, vanilla, and lemon juice and rind. Cover and cook over medium heat for 30 minutes, stirring occasionally. Uncover and cook until the potatoes are fork tender, about 10 minutes. Add the remaining butter and stir until completely melted. Cook uncovered until sauce is thick, about 2 minutes more.
Note: If fresh sweet potatoes are not available you may substitute 1 (15 ounce) can sweet potatoes packed in syrup. To use canned sweet potatoes, drain them, reserving 1/2 cup of the syrup. In a 1-quart saucepan place the reserved syrup, 1/3 cup sugar, 4 tablespoons unsalted butter, 1 tablespoon vanilla extract, and the juice and grated rind from 1/2 lemon. Cook over high heat for 2 minutes, whisking frequently. Add the sweet potatoes and reduce the heat to very low. Cover and simmer for 10 minutes. Remove the cover and cook until the mixture is reduced to 1 cup, about 10 minutes more, stirring occasionally and being careful not to break up the potato pieces.