For the Beer Braised Onions:
Peel and thinly slice the onions from top to bottom. Melt the butter in a wide sauté pan or rondeau over high heat. Add the onions, sugar and salt. Allow the onions to caramelize and develop browning in the bottom of the pan. As the caramelization begins to accumulate, deglaze the pan with about 1 ounce water at a time, just enough to moisten the caramelization and dissolve it into the onions. Repeat this caramelization and deglazing step until the onions have achieved a rich, deep mahogany brown. Add the beer, bring the mixture to a boil and then remove from the heat. The onions are now ready to use. You can substitute white wine for the beer if you prefer.
For the Pickled Pears:
Peel the pears, cut them in half from top to bottom, and put in a non-reactive bowl. In a stainless steel saucepot, bring the water, vinegar, sugar and salt to a boil, then pour this hot brine over the pears and allow them to cool in the brine. Refrigerate until you are ready to use them. This can be done several days ahead. Thinly slice the pears from top to bottom, so that they are thin enough to be pliable, about 3 millimeters thick.
For the Pickled Mustard Seeds:
In a stainless steel pot, combine the mustard seeds, water, vinegar, sugar and salt and simmer over low heat for about 45 minutes. Most of the liquid will be absorbed, and the mustard seeds will become plump and tender with a marvelous texture. You can store these in the refrigerator for months.
To Assemble and Serve:
Preheat the oven to 350°F. In a small 6-inch cast-iron skillet or earthenware cazuela, warm ¼ cup of the Beer Braised Onions. Top the onions with a piece of the grilled bread and several slices of the gruyère. Put the skillet in the oven for a few minutes to melt the cheese. Meanwhile, in a small sauté pan over a medium heat, sear a few hedgehog mushrooms and toss them with the sliced ham, some Pickled Asian Pears and some pickled mustard seeds. Arrange the mushroom mixture on top of the melted cheese.