In a large, oven proof skillet heat 1 tablespoon of butter over medium high heat. Add the morels and cook for 1 to 2 minutes, until they have released all their water and continue to cook until the water has reduced. Add the shallot and sauté for another 3 to 4 minutes, stirring often. Remove the vegetables from heat and set aside.
Wipe out the pan and add 2 tablespoons of butter to the pan.
Generously season the meat with salt and pepper. Then place the meat in the pan. It should sizzle on contact. Sear the meat until a brown crust forms (about 2-3 minutes each) on each side.
Remove from the stove top and place in the oven to cook for 15 to 20 minutes for rare to medium rare.
Remove from oven and let stand for 5 to 10 minutes.
While the meat rests return the pan to medium high heat. Add the final 2 tablespoons of butter and allow it to melt. Add the flour and stir well to make a smooth sauce. Deglaze the pan with the broth, thyme, and pepper and stir until well combined, scraping the browned bits from the bottom of the pan. Add the morel shallot mixture and any liquid back into the pan and cook for 1 to 2 minutes, or until the morels are heated through. Stir to combine and remove from heat.
Slice the meat and serve atop the morel gravy immediately.
Recipe by Sweet Love and Ginger at http://www.sweetloveandginger.com/venison-with-morel-sauce/