first off, adding a beaten egg is almost surely going to solve your problem. it'll make stuff stick. think of pancakes, that batter is pretty damn wet, but it still holds together when cooked. that's cause of the egg.
however, if the recipe doesn't call for egg, there should be a possibility of making it hold together without egg. I almost certainly think this has to do with the consistency of your batter. depending on the amounts of water (or milk etc) and oil (or butter) in your batter, it will look like the following:
- fluid, like pancake batter: lots of water
- sticky: high amount of water, some oil
- less sticky but still like paste: less water, more oil
- a bit sticky but comes loose from the bowl: slightly too much water
- not quite sticky and comes loose from the bowl: probably just the right amount of water and oil
- crumbly: not enough of either
- flaky or has little rolls in it: barely enough water or oil, add a bit more of both or either
- powder: dry
there's probably more states of batter. you can easily explore them by combining different amounts of olive oil, water and flour. if you start with small amounts and work until you get "not sticky + comes loose from the bowl" you end up with great pizza dough
[form a ball and let it rest for half an hour so the gluten can form their elastic bonds before using it]
I'm not sure if you can get this consistency with the chickpeas though, lacking flour. But check what the recipe asks for.
but, if like LMNO says, it's the oil temperature, there's another thing you gotta remember. if you throw in a refridgerated falafel slice, the temperature will drop very quickly, but only in the oil directly around the cold falafel bit. so the trick is to move the oil around a bit, to spread the temperature loss until it's back up to heat again. don't move the falafel pieces themselves, just carefully slush the hot oil around them a bit.
Also, get yourself a "blender-on-a-stick" (I forget what these things are called in English), they're not expensive and tremendously versatile, and will fullfill most of your foodprocessing needs, blend pastes, soups, butter, smoothies, pasta sauce and stuff like that. only thing they won't do is kneed dough, you need a mixer with those spirally dough curls for that. or your hands.