In a medium saucepan, combine the vegan butter, white sugar, brown sugar, almond milk, and cocoa powder. Place over medium heat and stir until everything is fully melted and smooth.
Bring the mixture to a boil. Once it begins boiling, let it cook for 1 minute, stirring continuously so the mixture doesn’t burn or stick to the bottom of the pan.
Remove the saucepan from heat and immediately stir in the peanut butter and vanilla extract. Mix until smooth and fully combined.
Add the quick oats and stir until all of the oats are evenly coated in the chocolate mixture.
Using a spoon or cookie scoop, drop portions of the mixture onto parchment paper. Gently shape the cookies if needed while they are still warm.
Let the cookies sit at room temperature until fully set and firm.