If you have large mushrooms, slice them a scant 1/4 inch thick, then cut the slices crosswise in half.
If you have smaller mushrooms, leave them whole or slice them in half.
What you are looking for is bite-size pieces.
Put a medium skillet over med-high heat and add the butter and oil.
When the bubbles from the butter have subsided, toss in the shallot and cook, stirring, until glistening and starting to soften, about 2 minutes.
Add the mushrooms, season with salt and pepper, and cook, stirring often, until the mushrooms have given up their liquid.
Continue cooking and stirring for another 2 minutes or so, until the mushrooms are tender, then pour in the cream.
Bring the cream to a boil and cook for 3 minutes or so, until it thickens just a little, then pull the pan from the heat and stir in the rosemary and mint.
Arrange the brioche on salad plates and top with the mushrooms and then the eggs.