If This Were Me: I would chop up the mushrooms but not so fine that you can't pick them out if you want to. Then I'd dump the stuffing into the widest bottom pan you have (for best surface area), mix the uncooked mushrooms in, then turn on the heat to medium. I'd want to mix everything thoroughly and cook until the mushrooms start looking squishy. You can tell when the oil is about to come BACK out of them when they start sweating/looking shiny. At that point stop. Take out the mushrooms if you want, or just eat around them.

I should say this has only worked for me once in a vegetable context, ie, they were far too greasy and the mushrooms basically wiped them up. I'm NOT sure how easily you can pull oil back out of bread, which is also absorbent in the same way that mushrooms are. Fingers crossed, if you try this tactic. XD