Step 1Melt 4 tablespoons of the butter in a 4-quart, heavy-bottomed stockpot or saucepan over low heat. Add the onions and garlic and cook over low heat for 10 to 15 minutes, or until the onions are translucent, stirring frequently. Add the chopped ham and continue to cook over low heat for another 10 to 15 minutes, stirring occasionally. Add the milk, cream and broth. Increase the heat to medium-high and continue to cook, stirring occasionally, until the soup begins to simmer. Reduce the heat to a gentle simmer, cover and continue to cook for 1 hour.
Step 2Remove from heat, strain the soup and reserve the liquid. Working in batches to avoid overloading, put the ham and onion mixture into either a blender (preferred) or food processer and add just enough of the reserved liquid to allow the ham to be processed by the machine. Process until the ham and onion mixture is finely ground, at least 5 minutes, and then gradually add more liquid until the ham looks like a smooth puree. Pour the ham puree back into a heavy-bottomed stockpot or saucepan. Repeat this process as necessary until all the ham has been blended. This makes about 5 cups soup.
Step 3Add any remaining liquid reserved from straining the soup, and blend with an immersion blender until the soup is one consistency. Season with salt and white pepper to taste.
Step 4To make the croutons: Heat the oven to 350 degrees. Trim the crust from the bread and cut the bread into rectangles about 1-by-1 1/2 inches that are one-fourth-inch thick; this makes about 24 croutons. Melt 3 tablespoons of the butter and toss the slices in the melted butter. Place the buttered slices of bread onto a cookie sheet and bake until golden brown, 3 to 5 minutes.
Step 5To make the brown butter: Place the remaining 4 tablespoons of butter in a heavy-bottomed pan. Heat over medium-high heat until the butter begins to brown. Remove from heat immediately. Set aside.
Step 6To plate the soup: Cut each slice of cheese into rectangles similar in size to the croutons. Place slices of cheese on top of the croutons and bake in a 350-degree oven until the cheese has melted, about 2 to 3 minutes. Fill a soup bowl with desired amount of soup. Place the croutons with melted cheese on top of the soup. Drizzle a little brown butter over and around the croutons. Sprinkle julienned cornichon and radish on top of the croutons. Either pour the beer into a glass from a distance to create a large head on the beer, or blend the beer to create foam. Spoon the beer foam on top of the soup. Serve immediately.