Place the cubed bread on a sheet pan and toast in a 350° F oven until somewhat dried (~5 mins) but not brown.
Set a 10-inch skillet over medium-high heat and add the olive oil to the pan; add the chopped onion and a pinch of salt. Cook until the onion begins to soften; add the minced garlic and stir until it is fragrant, then add the chopped mushrooms. Cook and stir until the mushrooms brown and give off some liquid.
Add the vermouth and cook until the liquid reduces to a glaze. Remove from the heat and set aside.
Lightly oil a 4-cup casserole or gratin dish. In a medium bowl, beat the eggs together with the cream and chicken stock. Add 1/2 cup of the shredded cheese. Fold the dried bread cubes and the mushroom mixture into the custard, pressing the bread down into the liquid. Let stand while you preheat the oven.
Preheat the oven to 350° F. Place a rack in the lower third of the oven. When the oven is heated, transfer the bread and vegetable mixture to the prepared baking dish. Press the bread down into the custard and smooth the top surface a little; sprinkle on the remaining 1/4 cup shredded cheese.
Place in the oven and bake for 30 to 45 minutes, or until the top is golden brown and the custard is set. The baking time will depend on the depth of the baking dish you have chosen. Leftovers can be refrigerated, tightly covered for 2 to 3 days.