You don't have to "ferment" them (note: that's not fermentation, just dissolution, and it doesn't require frying, although it might speed up the process by oxidizing CaCO3 to CaO if hot enough and/or burning off the membrane). But if you want to add calcium to your soil, eggshells are a calcium concentrate
Eggshells are basically calcium carbonate plated out around a thin organic membrane. Vinegar is a weak acid and dissolves the calcium carbonate, akin to acid rain dissolving limestone (which is also calcium carbonate).
Dissolving with eggshells with vinegar will create calcium acetate. Honestly, there are better anions than acetate for supplementing soil nutrients (sulfates, phosphates, nitrates...), and you definitely don't want residual acetic acid (would you dump plain vinegar on your soil?
), but it's certainly workable, and has a history of use. Soluble forms of calcium of course go straight to work in the soil. However, calcium carbonate itself is active in the soil - that's what agricultural lime is. It just takes longer to act. Also, it reduces soil acidity, while calcium acetate should be pH neutral. But the effects are not going to be huge because you're not going to be using huge amounts. If you're just talking kitchen waste, it's not going to be that much calcium, and even less effect on pH.
Doing some quick checking I see that the most desirable use for acetate fertilizers is for foliar sprays, due to their high solubility. But again, you definitely don't want to be dumping residual vinegar on your plants.... unless you also are dumping some olive oil and calling it a salad
If you decide to dissolve eggshells in vinegar, make sure it's the *eggshells* that are in excess, not the vinegar! If you want to speed up the reaction, in addition to grinding the eggshells as fine as you can, and heating as hot as you can get them, having the dissolution occur in hot vinegar will greatly speed up the process (and help make it more thorough). When I'm testing mineral samples for their calcium content, I always use hot vinegar rather than cold; cold will dissolve small calcite crystals, but its a much more vigorous, obvious reaction with hot vinegar. Just be aware that hot vinegar is... well, pungent!
