Step 1Heat the oven to 375 degrees.
Step 2Heat a saute pan over medium-high heat. Add the olive oil, then the mushrooms and a pinch of salt and saute until the mushrooms are softened, 5 to 7 minutes. Stir in the garlic, reduce the heat to medium and continue to cook, stirring constantly, until the garlic is aromatic, 1 to 2 minutes more. Remove from heat and place the mushrooms in a medium bowl to cool.
Step 3When the mushrooms are cool, stir in the spinach and chicken. Add the cream cheese, gently tossing in almond-sized pieces, then gently stir in the shredded cheese and season to taste with salt and pepper. If desired, stir in the egg to use as a binder.
Step 4Layer 5 sheets of phyllo one directly on top of another, lightly brushing each with olive oil. Place half of the chicken filling in a log shape along the short end of the phyllo rectangle. Roll the filling in the phyllo, similar to a burrito, tucking in the edges as you go to prevent the filling from spilling out. Brush the top of the strudel lightly with olive oil and place on a parchment-lined baking sheet. Repeat with the remaining 5 sheets of phyllo and filling.
Step 5Bake the strudel until lightly golden-brown on top and warmed through, 15 to 20 minutes. Cool slightly before slicing.
Step 1Heat a large saute pan over medium-high heat. Add the olive oil and tomatoes and saute, stirring frequently, just until they start to break down, 5 to 7 minutes. Stir in the garlic and reduce the heat to medium. Simmer the mixture, stirring occasionally, until the garlic softens and the sauce begins to thicken, about 10 minutes.
Step 2Stir in the cream and season with 1 teaspoon, or to taste, of salt and several grinds of fresh pepper. Stir in the basil and thin, if desired, with white wine. Remove from heat.
Step 3Slice the strudel into wedges and serve with the tomato compote.