Cut the limes into fourths and squeeze out the lime juice directly into the glass and over the brown sugar. Swirl the juice and sugar once in a while as you prepare the rest of the recipe so that the sugar can dissolve.
Meanwhile, peel the ginger and cut thinly into slices and then into thin strips/pieces.
Rinse the mint. Squeeze some of the mint leaves (or even leaves with a stem) and add to the glass with the sugar lime syrup. Swirl a little.
Add the ginger pieces and some ice cubes and the squeezed out lime quarters (they still have enough juice in them). I like to arrange the ice and lime quarters in an alternating pattern so it's not all in one place.
Pour ginger beer into the lemonade glass until full (the amount will vary depending on the amount of ice used). Stir a bit in order to mix the lime syrup with the ginger beer. Otherwise, all the sweet syrup will be on the bottom and the rest of the lemonade will be less sweet.
Insert a mint stem into the glass for garnish.
Serve immediately.
Notes
Note if your drinking glasses are on the bigger side, you might need to double the amount of the drink. My glasses fit about a cup of ginger beer once the ice, lime, mint, and ginger are added.