What's this? A skim of the fingers and we have biochar!!
"Biochar enthusiasts generally agree that raw biochar needs to be processed further prior to being added to the garden. Composting, or soaking with compost tea is commonly used to charge the pore volume with beneficial organisms and nutrients. Soaking in a nutrient rich solution (examples are urine or fish emulsion) prior to composting is accepted practice." from a Gardening with Biochar FAQ.
The largest pieces will float to the top and can be skimmed off... there will still be some worth saving at the bottom of the bucket amidst the ash.
Remember that folks used to make soap out of lye which was made by dripping water through wood ash (particularly oak) and fat.
Frankly, biochar is just 'charred but unburned' organic matter. Fireplaces won't be too good at 'making' it, but inserts and woodstoves will. What you want is 'incompletely' burned wood which means making the air mix just a little lean. Smoldering fire is what you are looking for. Yes, you can spend good money on 'biochar', although I'm having trouble googling any sources just now. But do your research and think about it.
We burn wood anyway, so we'll be thinking more about retrieving those bigger 'charred' logs that used to get pushed back in to burn more completely. We have been fortunate to be able to 'salvage' the trees we use for wood and this is one more way to use them!