Add the first five ingredients (through the salt) to a 4-quart saucepan. Bring to a rapid boil and let boil for 1 minute. Remove from heat. Stir in the peanut butter and vanilla until smooth, then stir in the oats.

Using a medium cookie scoop, drop onto parchment-lined baking sheets. Let cool until set, about 30 minutes. Cookies can be stored in an airtight container at room temperature or in the refrigerator for up to 2 weeks.