Combine peaches and lime juice in a blender. Puree until very smooth.
Pour into a saucepan over medium heat, stir in 1/2 cup sugar, and bring to a boil. Cook, stirring continuously, until thickened, about 15 minutes.
Stir in remaining 1 1/2 cups sugar and pectin. Using a thermometer, heat to 205 degrees F (96 degrees C) and cook, stirring continuously, for another 10 minutes. Remove from heat.
Pour peach puree into the prepared baking dish. Shake gently and tap on the countertop to remove any air bubbles. Cover and refrigerate at least 8 hours or overnight.
Sprinkle about half the 1/2 cup of sugar over a silicone baking mat and invert the peach gelee on top. Remove plastic wrap and sprinkle top with sugar. Trim off any uneven edges and cut gelee into 25 squares.