Preheat oven to 350.
Preheat a large, heavy-bottomed saucepan over medium heat. Melt butter and add onions and garlic. Season well with sea salt and freshly ground black pepper and sautee until onions are translucent and starting to brown. Deglaze pan with the wine and cook until the alcohol smell dissipates. Add the chicken stock and water and then whisk in the soups until smooth. Return to a boil and then add the rice. Reduce heat to simmer, cover and cook about half an hour, stirring occasionally, until the sauce thickens and the rice is partially cooked.
While the sauce is simmering, wash the broccoli heads, remove the florets and cut into uniformly sized pieces, trimming off any long stems. Peel the broccoli stems, cut into half-inch cubes and add them to the prepared florets. Shred the cheeses and toss thoroughly to combine.
When the sauce has finished simmering, remove from heat and stir in 3/4 of the cheese mixture, retaining half a cup. Stir until smooth and check the seasoning, adding more salt and/or pepper if needed. Add the prepared broccoli and stir to combine. Turn the mixture into a three inch deep 9x9 casserole and spread evenly. Top with the remaining cheese and bake, uncovered, for one hour until the rice is cooked, sauce is thickened, broccoli is tender and cheese is browned. Let stand 10-15 minutes before serving.