1 tablespoon extra-virgin olive oil
6 tablespoons salted butter, divided
2 cloves garlic, finely chopped
2-10 ounce packages frozen chopped broccoli, thawed
2 large eggs
16 ounces cheddar cheese, shredded (4 cups)
2 teaspoons Dijon mustard
10 ounce sheet phyllo dough, thawed
Preheat oven to 375F. Line a baking sheet with parchment paper and set aside. In a large skillet, heat olive oil and 2 tablespoons butter over medium heat. Add garlic and cook 5 minutes. Stir in broccoli and remove from heat. Transfer 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 broccoli mixture along with cheddar cheese and mustard. Set aside.
In the skillet, melt the remaining 4 tablespoons butter.
In a small bowl, lightly beat remaining egg and set aside.
Place 1 sheet phyllo dough on a work surface with the short side closest to you; cover remaining phyllo sheets with a damp kitchen towel. Brush phyllo sheet lightly with melted butter, beginning with the edges. Stack 4 more phyllo sheets on top, brushing all but the last sheet with butter. Place half of reserved broccoli mixture on short end of the phyllo, leaving a 3-inch border of dough along sides and bottom. Fold bottom edge over filling, then fold in the two sides and gently roll up into a log, brushing inside flap with egg to seal the strudel. Repeat with remaining ingredients to make another strudel.
Place strudels seam side down and 3 inches apart on prepared baking sheet. Brush tops and sides with remaining egg. Bake until golden-brown, 30 to 35 minutes.