Sauté shallots for the ragù 1 T. oil in a large sauté pan over medium heat until soft. Increase heat to high, add half the mushrooms, and sauté until browned; remove from pan. Sauté mushrooms in batches so they brown well. Resist stirring them too much, the longer the contact with the pan, the better the browning. Add additional tablespoon of oil to the pan and sauté remaining mushrooms, then return the reserved mushrooms to the pan.
Add tomatoes, broth, sherry, vinegar, and tomato paste. Reduce heat to low and simmer for 15 minutes. Finish with herbs and seasonings. While the ragù simmers, make the polenta.
Bring broth, cream, and seasonings to a boil for the polenta in a saucepan. Whisk in cornmeal; reduce heat to low and simmer 5 minutes, stirring often. Fold in cheeses before serving.
Place polenta on plate, topped with mushrooms ragu.