Preheat the oven to 375 degrees . Line a baking sheet with parchment paper and set aside. In a large skillet, heat the olive oil and 2 tablespoons butter over medium heat. Add the leeks and garlic and cook until the leeks begin to brown, 5 to 7 minutes. Stir in the broccoli, season with salt and pepper and remove from the heat. Transfer the broccoli mixture to a large bowl and let cool for 10 minutes. Wipe out the skillet and set aside.
In a small bowl, lightly beat 1 egg and stir into the broccoli mixture along with the cheddar cheese and mustard. Set aside.
In the skillet, melt the remaining 4 tablespoons butter. In a small bowl, lightly beat the remaining egg and set aside. Place 1 sheet phyllo dough on a work surface with the short side closest to you; cover the remaining phyllo sheets with a damp kitchen towel. Brush the phyllo sheet lightly with the melted butter, beginning with the edges. Stack 4 more phyllo sheets on top, brushing all but the last sheet with butter. Place half of the reserved broccoli mixture on the short end of the phyllo, leaving a 3-inch border of dough along the sides and bottom. Fold the bottom edge over the filling, then fold in the two sides and gently roll up into a log, brushing the inside flap with the egg to seal the strudel. Repeat with the remaining ingredients to make another strudel.
Place the strudels seam side down and 3 inches apart on the prepared baking sheet. Brush the tops and sides with the remaining egg. Bake until golden-brown, 30 to 35 minutes.