Peel the fresh ginger root and chop it finely or grate it to release its juices. Measure out 1 cup of prepared ginger.
In a medium saucepan, combine the chopped ginger and 1 cup of granulated sugar. Add 2 cups of filtered water to the saucepan and bring the mixture to a boil over medium heat, stirring occasionally until the sugar is completely dissolved. Once boiling, reduce heat and let it simmer for 10 minutes.
Remove the saucepan from heat and allow the ginger syrup to cool down to room temperature. Once cooled, strain out the ginger pieces using a fine mesh sieve or cheesecloth, and set the syrup aside.
In a large mixing bowl or pitcher, combine the cooled ginger syrup, remaining 2 cups of filtered water, and 1 cup of fresh lemon juice. Stir well to mix all the ingredients together.
In a small bowl, dissolve 1 teaspoon of active dry yeast in a few tablespoons of warm water (not boiling) and let it sit for about 5-10 minutes until it becomes frothy.
Add the activated yeast to the ginger mixture and stir gently to combine. Pour the mixture into clean bottles, leaving some space at the top of each bottle for carbonation. Seal the bottles tightly.
Leave the bottles at room temperature for 24-48 hours to allow fermentation to occur. Check the bottles periodically for carbonation by gently squeezing them; they should feel firm when ready.
Once the desired carbonation is reached, place the bottles in the refrigerator to slow down the fermentation process.
Pour the ginger beer into glasses over ice. Garnish with lemon slices and fresh mint leaves to taste.