Combine the morels and water in a large mixing bowl and let soak at room temperature for 2 hours.
Heat the oil in a large saucepan over medium heat.
Add the shallot to the pan. Cook, stirring occasionally, until tender and without any color, about 4 minutes.
Add the sliced mushrooms and cook, stirring occasionally, until the mushrooms are caramelized and the liquid released from the mushrooms is fully evaporated, about 8 minutes.
Add the sherry to deglaze and scrape the caramelized bits from the bottom of the pan.
Bring the wine to a simmer and reduce until almost dry.
Add the rehydrated morels with the soaking water.
Bring the liquid to a simmer and add the thyme. Simmer the mixture for 20 minutes.
Remove the pan from the heat and strain through a chinois.
Heat a water bath to 62°C/145°F.
Return the mixture to a clean saucepan and heat to a simmer over medium heat.
Continue to simmer until reduced by half, to about 100 g ( ~1/2 cup). Crack the egg and egg yolks into a mixing bowl and whisk to combine.
Whisk the warm mushroom base into the eggs, being careful not to scramble the eggs.
Once combined, slowly whisk in the melted butter, vin jaune, and vinegar.
Season the mixture with salt.
Transfer the sabayon to an iSi canister and charge with two charges of N2O.
Cook the sabayon in the water bath for 45 minutes.
Keep warm.