Preheat the Oven: Preheat your oven to 350°F (175°C). Line a large cookie sheet with parchment paper or a silicone baking mat.
Cream the Butter and Sugar: Soften the butter slightly in the microwave for 10-12 seconds. In a large mixing bowl, combine the softened butter with the granulated sugar. Beat on medium speed until the mixture is well creamed together, light, and fluffy.
Add Eggs and Vanilla: Add the eggs and vanilla extract to the butter and sugar mixture. Mix on medium speed until the eggs are fully incorporated and the mixture is smooth.
Mix Dry Ingredients: In a separate bowl, whisk together the flour, baking soda, cream of tartar, and salt. Gradually add the dry ingredients to the wet ingredients, mixing on medium-low speed until a dough forms. Fold in the Fruity Pebbles cereal until evenly distributed throughout the dough.
Portion and Bake: Use a large cookie scoop to portion the dough into balls. Roll each ball of dough in granulated sugar, then place them on the prepared baking sheet, spacing them about 2 inches apart. Bake in the preheated oven for 10-12 minutes, or until the edges are lightly golden. Allow the cookies to cool on the baking sheet for several minutes before transferring them to a wire rack to cool completely.
Coat with White Chocolate: Break the white chocolate bar into smaller pieces and place them in a microwave-safe bowl. Microwave at 50% power in 30-second intervals, stirring after each interval, until the chocolate is fully melted and smooth. Dip the cooled cookies into the melted white chocolate, tapping off any excess chocolate. Place the dipped cookies on a sheet of parchment paper to cool and set.