Preheat oven to 425 degrees F (220 degrees C). Oil a large baking sheet.
Stir sugar, soy milk, peanut butter, canola oil, and vanilla extract together with a whisk in a large bowl until completely smooth.
Mix flour, oats, baking soda, and salt in a separate bowl; add to the peanut butter mixture and stir to combine. Fold chocolate chips and walnut pieces into the flour mixture.
Drop your batter by large spoonfuls onto prepared baking sheet.
Bake cookies in preheated oven until browned along the edges, about 10 minutes. Cool cookies on sheet for 10 minutes before removing to a cooling rack to cool completely.