Watching This Old House, they showed a method for joining wood trim using a two part adhesive that bonded instantly within seconds. Glue was applied to one side while an accelerator was sprayed on the other. Researching online, I found the product is just cyanoacrylate (CA) commonly known by the brand name Crazy Glue used with an accelerator. CA accelerator has several brands including Zip Kicker, Instant Bond and Insta-set. Since it bonds almost instantly, it is stronger and there is no waiting for the product to dry.
I have been buying crazy glue at the dollar store since you can get mini tubes and don’t need to waste as much when the whole tube dries out. There are some videos on youtube where baking soda is used as an accelerator as well. Here is an article with the Mythbusters about it.
Zip Kicker can be found at hobby shops (popular with model builders) and online. I’ll have to try it out next time I break something.