1. Make filling: 
Heat olive oil in a large skillet over medium heat. Add ground American lamb and cook until browned and starting to crisp, breaking it apart with a spoon, about 5–7 minutes. Transfer to a bowl and set aside. 

In the same pan, add onion and garlic. Sauté until softened, about 3 minutes. Add the mushrooms and cook until they release their liquid and it evaporates, about 8–10 minutes. Stir in tomato paste and thyme; cook for 1 minute to caramelize slightly. Deglaze with balsamic vinegar and red wine, scraping up any browned bits. Simmer until most of the liquid has evaporated. 

Return the lamb to the pan, stir in butter and flour, and cook another 1–2 minutes until the mixture thickens and coats the lamb. Season with salt and pepper to taste. 

Remove from heat, stir in parsley and shredded gruyere, and let the filling cool to room temperature in the fridge. 

2. Prepare filo: 
Preheat the oven to 375°F and line a baking sheet with parchment paper. 

Unroll the filo dough and cover it with a slightly damp kitchen towel to keep it from drying out.

Lay one sheet on a clean surface and brush lightly with melted butter. Top with another sheet and brush again. Cut the stacked sheets lengthwise into four long strips (about 3 inches wide). 

3. Fold triangles: 
Place about 2 tablespoons of lamb-mushroom filling at one end of each strip. Fold one corner over the filling to create a triangle, then continue folding it up the strip like a flag to encase the filling completely. Place the finished triangles seam-side down on the prepared baking sheet and brush the tops with melted butter. 

4. Bake: 
Bake for 18-20 minutes, until golden brown and crisp. Let cool slightly before serving.