How can I thicken my mushroom sauce?
You could use any number of methods, including:


Starch Thickeners (added as a slurry)

One of the simplest and most straight forward:  dissolve some starch (cornstarch is common in the US; alternates include potato starch, arrowroot, or tapioca) in some water, into a smooth slurry.  

Add the slurry to your simmering sauce base, stirring, and let it cook for a minute or so.  As long as your sauce is at the simmer, it will come to full thickness very quickly.

Add slowly until the desired thickness is reached.
Flour slurry

The technique is similar to using a pure starch:  create a slurry of flour in water, enough water so that it will pour easily.

However, you will want the sauce base at a boil, as flour thickens at higher temperatures.  It will also add more cloudiness to your sauce (probably not a factor with your recipe).

Add slowly until the desired thickness is reached.
With a beurre manee (butter kneaded with flour)

Knead equal parts of room temperature butter and flour together.

Add slowly to your boiling sauce, while whisking, until the desired thickness is reached.
With roux (flour cooked with butter)

Melt butter in a saucepan.  Add an equal volume of flour, whisking it in to prevent lumps.  Continue cooking for a couple minutes.  You can use this right away, or save it in the refrigerator to use later.

Add roux to your boiling sauce base, whisking, until you achieve your desired thickness.


All of these starch based methods will thicken about one to two cups of sauce per tablespoon of starch, depending on how much additional thickness you desire.  The exact ratio of thickener to sauce is up to your taste.

All of the starch based methods should be done at the end of the cooking period, prior to service.  


Reduction

Simply reduce your sauce until it is as thick as you desire.  This may overcook your mushrooms so may not be an ideal method for your recipe.  You might also reduce your sauce to almost the desired thickness, then add the mushrooms.  Your sauce is very, very high in fat (butter and marscapone, so it will not likely break).
More mushrooms (and a blender)

Add even more mushrooms to your recipe.  When it is cooked, remove some mushrooms and reserve them.  Puree the remaining mushrooms and sauce in a blender or with an immersion blender (or even with a food mill).  Add the whole mushrooms back for flavor.  The body of the pureed mushrooms will add thickness to your sauce. 

This method will work even better if you add some potato chunks with your mushrooms.  This gives the benefits of the potato starch, and more body from the potatoes when you puree.