Preheat the oven to 350°F / 175°C. Line a muffin tin with 6 parchment liners.
To a medium sized mixing bowl, add the granulated sugar. Zest the lemon into the bowl, then use your fingers to rub the zest and sugar together until the mixture resembles wet sand.
Add in the flour, baking powder, salt and cubed butter. Use a hand mixer or stand mixer fitted with the paddle attachment to beat the mixture together until the butter has completely combined with the dry ingredients and no lumps remain.
Add in the oil, egg white, lavender extract, and buttermilk. Whisk the ingredients together just until a cohesive batter has formed.
Distribute the batter evenly between the six liners. Bake for 15-18 min., or until a toothpick inserted in the center of the cupcakes comes out clean.
Lemon Syrup
Juice the lemon you zested to get ~25g/2 tbsp. of juice (it's okay if it's a little less - just add an equivalent amount of sugar.) Add in the sugar, then microwave in 20-second intervals, stirring after each, until the sugar has dissolved.
Poke some holes in the tops of each cupcake with a toothpick, then brush some of the lemon syrup over them.
Lavender Buttercream
In a medium sized mixing bowl, beat the butter for 1 minute to remove any lumps.
Slowly beat in the powdered sugar until all of it has been used.
Add in the lavender extract, heavy cream, and purple food coloring (if using). Beat for an additional minute.
Once the cupcakes have completely cooled to room temperature, transfer the frosting to a piping bag and frost each cupcake. Garnish with fresh lavender and lemon slices if desired. Serve and enjoy!