Combine 1 1/2 cups plus 2 1/2 tablespoons water, 2 teaspoons sugar, 2 teaspoons ground ginger, and 1 3/4 plus 1/8 teaspoons yeast in a container. Mix together to make the bug. Cover loosely and let stand in a warm place for 24 hours.
"Feed" the bug daily for 7 days by adding 1 teaspoon sugar and 1 teaspoon ground ginger.
Strain the bug through muslin cloth, pouring the liquid into a separate large container.
Mix the strained liquid with 3 quarts cold water. Add lemon juice to the brew.
Dissolve 3 1/4 cups plus 2 1/4 tablespoons sugar in boiling water; add to the brew. Mix in grated ginger. Bottle the ginger beer. Let stand until fizzy, at least 1 week. Refrigerate before drinking.