Prepare your pressure canner according to the manufacturer instructions and bring clean pint jars to a simmer to keep them hot.
In a large stockpot, melt the butter over medium heat. Add the onions and garlic and saute until the onions are translucent and soft, about 15 minutes.
Add the sliced mushrooms and saute for another 5 to 10 minutes, until they begin to brown.
Add the wine, broth, salt, pepper, and thyme. Bring to a boil and boil for 5 minutes to heat everything through and drive air from the mushroom tissue.
Ladle the solids into hot pint jars first, dividing them evenly, then top each jar with the hot liquid, leaving 1 inch of headspace.
Remove air bubbles, recheck headspace, and add boiling water or broth to return to 1 inch if needed.
Wipe the jar rims, center the lids, and screw on bands until fingertip tight.
Vent the canner for 10 minutes, then process pints at 10 lbs (weighted) or 11 lbs (dial) for 45 minutes. Adjust pressure for your altitude. Process pints only; there is no tested process for quarts.
Turn off the heat and allow the canner to depressurize naturally. Wait until fully depressurized before opening.
Remove the jars and cool undisturbed for 12 hours. Check the seals, label, and store sealed jars up to 12 months.