Lightly grease an 8X8 baking dish with olive oil or butter.
In a large saucepan, combine chicken stock, milk and bay leaf. Bring to gentle boil, then cover, remove from heat and allow to sit for 10-15 minutes.
Remove bay leaf. Bring to a boil, then reduce to medium and slowly sprinkle in cornmeal, vigorously whisking in each portion until well-combined. Cook over low heat about 5 minutes, stirring with a wood spoon or heat-proof spatula, until it's smooth. Remove from heat and stir in butter, cheese, salt and pepper.
Pour polenta into the baking dish and let it sit at room temperature, uncovered, ~1 hour, until it firms up. (If planning to use later, place in the refrigerator after it cools).
When ready to prepare, cut the polenta into squares. Heat a non-stick pan over medium/high heat, along with a little butter and olive oil. Place polenta in pan and reduce heat to medium. Cook until first side is nicely browned, then flip and cook until it's also brown to your liking. Serve immediately with Balsamic mushrooms.