Lees in the bottle after carbing?

Last December I brewed a cider and bottled it after "a few days" of fermentation (that's as specific as my notes go - this was only my second batch, give me a break) while it was still going strong. I tried to avoid getting the lees in my carboy into the bottle, but now that I'm opening them to drink them I'm getting a TON of carbonation (it takes about 20 minutes to open a single bottle, since I crack the lid a bit to vent pressure and then let it settle, and repeat the process) and, of more concern, I've got a lot of crap on the bottom of the bottle.

Obviously letting it ferment longer before bottling will reduce both of these problems, but I've found that almost all of my brews have some amount of stuff settled out in the bottom of the bottles. Is this inevitable, or is there some way to get rid of it?

P.S. this batch was done with Nottingham Ale and it's fantastic - both carbed (though excessively so) and sweet, without pasteurization. Definitely going to try a larger batch with the same basic method.

Letting your cider clear in secondary, racking off the lees, and then bottling will reduce the lions share of sediment with minimal effort. To take this method further use pectic enzyme as well as fining agents if needed.

To eliminate lees completely, look up how champagne is made with special attention to riddling, disgorging, and dosage. This is a lot more work however.

Also, when planning to bottle condition, choose a yeast which will form compact lees.

if you are bottling while the fermentation is still active then the cider still has millions of yeast in suspension and that is what is dropping out to create lots of lees. If you let it ferment out and clear you can then back-sweeten/prime, bottle, carbonate then PASTEURIZE. This will give you a more manageable co2 level. Also it will reduce the lees as you will only have the small amount needed to carbonate. There will always be lees with bottle carbonation but by letting the cider clear first you will have much less.

Bottle sediment isn't a bad thing, but everyone is correct in saying that racking will clear quite a bit of it. From my experience, Nottingham Yeast seems to create a more compact bottle sediment where S-04 seems to be more 'chunky' and stirs up more.

Anyone else have sediment comparisons on these two?