Line a small baking sheet with parchment paper or a silicone mat. Set aside.
In a food processor, combine the figs, almond butter and water. Blend until smooth, scraping down the side of the bowl as needed with a rubber spatula. Using a small ice-cream scoop, scoop 1 tablespoon of the fig mixture at a time and roll into 1 1/2-inch balls. Insert a lollipop stick into each fig ball.
Place the chocolate chips and 1 teaspoon of the oil in a medium bowl. Place the bowl over a small saucepan of barely simmering water and stir until the chocolate is melted and smooth, about 2 to 3 minutes.
Drop the fig balls into the melted chocolate and, using a fork, roll to coat evenly. Remove the fig balls, allowing any excess chocolate to drip back into the bowl, and place on the baking sheet. Refrigerate for 1 hour, or until the chocolate has set.
Using a small pair of scissors, cut the fruit leather into small squares to make noses for the cats and triangles for the ears. Dip the ends of the ears in the leftover dark chocolate and stick them on the cats. Repeat with the squares for the noses.
Place the white chocolate and remaining 1/2 teaspoon of the oil in a medium bowl. Place the bowl over a small saucepan of barely simmering water and stir until the chocolate is melted and smooth, about 2 to 3 minutes. Cool slightly. Transfer the chocolate to a piping bag fitted with a small plain tip. Pipe eyes and whiskers on the cats. Refrigerate until set, about 30 minutes.