Preheat oven to 375°F (190°C) and line a circular baking stone or large baking sheet with parchment paper.
Heat olive oil in a large skillet over medium-high heat. Add onions, bell peppers, and mushrooms; sauté until softened and slightly caramelized, about 6 minutes.
Add the sliced ribeye to the skillet, cooking until just browned; season with salt and pepper and remove from heat.
Unroll crescent dough and separate into 16 triangles.
Arrange the triangles in a circle on the baking sheet with the wide bases overlapping in the center and the points facing outward, forming a sun-like shape with a 5-inch diameter hole in the center.
Layer the provolone cheese slices around the inner circle of the dough bases.
Spoon the beef and vegetable mixture evenly over the cheese.
Fold the pointed ends of the dough triangles over the filling, tucking the tips under the center edge to secure the ring.
Brush the exterior of the dough with the beaten egg wash.
Bake for 18 to 20 minutes until the crust is golden brown and the cheese is melted.
Cool for 5 minutes before slicing into wedges.