Preheat the oven to 350 degrees F.
In a medium, straight-sided saucepan over medium heat, melt 3 tablespoons of the butter.
Add the leeks and the garlic to the butter and cook, stirring occasionally, until the leeks have softened, about 4 minutes.
Add the flour to the leek mixture and cook, whisking constantly, for 3 minutes.
Whisk the heavy cream into the leek mixture until smooth.
Add the spinach to the leek mixture and cook, stirring occasionally, until it is very thick, about 5 minutes.
Stir the cheese, the lemon juice, the nutmeg, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper into the spinach mixture and transfer the saucepan away from the heat.
Let the spinach mixture cool for 10 minutes.
While the spinach mixture is cooling, use a sharp knife to make a horizontal cut through the center of the beef, cutting to, but not through, the other side. The cut should be within 1/2-inch of the other side.
Open the top cut piece (as you would a book) and lay it flat.
Place 1 sheet of plastic wrap on a flat surface and place the meat on top. Top the meat with another piece of plastic wrap.
Using the flat side of a meat mallet or the back of a small, heavy skillet, pound the meat to 1/2-inch thickness and into a 13-inch square.
Remove the plastic wrap and spread the spinach mixture evenly over the beef, leaving a 1 1/2-inch border around the edges.
Roll the beef and tie with kitchen twine, securing it at 2-inch intervals.
Brush the meat evenly with 1 tablespoon of the olive oil and sprinkle with 1 teaspoon of the salt and 1/2 teaspoon of the pepper.
Place the rolled beef on a wire rack and set it over a rimmed baking pan.
Bake the beef for 30 minutes.
Increase the oven temperature to 450 degrees F and bake until the beef's internal temperature reaches 135 degrees F, about 20 minutes.
Transfer the meat from the oven and let it stand for 30 minutes.
While the meat rests, melt 1 tablespoon of the butter with the remaining oil in a large skillet over medium-high heat.
Add the mushrooms to the butter mixture and cook, stirring once or twice, until they are lightly browned, about 5-6 minutes.
Add the shallot to the mushrooms and cook, stirring constantly, until the shallot is tender, about 1-2 minutes.
Stir the wine into the mushroom mixture and cook, stirring constantly, for 2 minutes.
Stir the broth into the wine mixture, then lower the heat to medium and simmer, stirring constantly, for 5 minutes.
Stir the mustard, the rosemary, the remaining salt, and the remaining pepper into the sauce mixture and cook for 1 minute.
Transfer the sauce from the heat and stir in the remaining butter until combined.
Serve the beef with the sauce.