Adjust oven rack to middle position and heat oven to 350 degrees. Line 2 rimmed baking sheets with parchment paper. Whisk flour, baking powder, baking soda, and salt together in medium bowl; set aside.
Place 1 cup Lavender Sugar, 1/2 cup sugar and cream cheese in large bowl.
Place remaining 1/3 cup sugar in shallow dish and set aside.
Pour melted butter over sugar and cream cheese and whisk to combine(some small lumps of cream cheese will remain but will smooth out later). Whisk in oil until incorporated.
Add egg, milk, and vanilla and whisk until smooth.
Add flour mixture and mix with rubber spatula until soft, homogeneous dough forms.
Divide dough into 24 equal pieces, about 2 tablespoons each. Using your hands, roll dough into balls. Working in batches, roll balls in reserved sugar to coat and evenly space on prepared sheets, 12 balls per sheet. Using bottom of drinking glass,flatten balls to 2 inches in diameter.
Sprinkle tops of cookies evenly with 4 teaspoons of sugar remaining in shallow dish (2 teaspoons per sheet); discard any remaining sugar.
Bake cookies 1 sheet at a time until edges are set and just beginning to brown, 11 to 13 minutes,rotating sheet halfway through baking. Let cookies cool on sheet for 5 minutes,then transfer to wire rack. Let cookies cool completely before serving.
Lemon Glaze (optional)
Mix lemon juice with enough powdered sugar to make glaze and drizzle over the cooled cookies - allow glaze to set.
The final dough will be slightly softer than most cookie doughs. For the best results, handle the dough as briefly and gently as possible when shaping the cookies. Overworking the dough will result in flatter cookies.