There is cheese sauce, and there is cheese sauce.  This is one of the best. However, that means a bit more work.

Add the milk, bay leaf, onion, clove and nutmeg to a small sauce pan and simmer over low heat, but don't let come to a boil,  for 15 minutes.

In another medium sauce pan, add 2 tablespoons of the butter.  Melt over a low heat. Once melted, add flour and stir until smooth.  Cook over medium low heat until a light golden brown, about 6 minutes, stirring occasionally.

Remove the onion, bay leaf and clove from the milk.  Slowly integrate the milk into the butter and flour (roux) by adding small amounts of milk and whisking into the roux. Repeat until all the milk is incorporated.  But go slow or you wind up with lumps.

To this (Béchamel sauce), slowly stir in the cheeses.   Reduce heat to low and stir occasionally.

In a saute pan, saute the mushrooms over medium low heat in the remaining butter until tender, about 10 to 15 minutes.

These are just square patties with salt and pepper to taste.  I made them square to fit on the square buns.  If you have round buns,  round patties.

On a prepared grill, over high direct heat, place the patties and grill for 4 to 6 minutes per side (depending on thickness) until cooked through.

When you flip, add a tomato slice on each burger to let the tomato cook.

Since this is poultry, if you are unsure of doneness, best to observe the rules of cooking temps for poultry and make sure internal temp is 160º F.

Slice 'em.

Toast them on the grill.  Watch 'em closely, they can burn real quick.

Follow the directions for cooking bacon.

In a baking dish, lay out the toasted bun bottoms.  Add the burger with tomato on top, one on each bun.  Add the mushrooms.  Add two slices of bacon.  Pour a good sized glob of sauce all over each burger.

Top with the bun, or serve open faced as typical Hot Browns are served.