Brush mushrooms clean with a damp paper towel. Brush with olive oil and season with salt and pepper. Place the mushrooms over Direct Medium heat and grill 10 to 15 minutes or until tender. Let cool; discard base of oyster mushrooms just where stems attach to each other. Chop mushrooms roughly. Place in a strainer and allow any excess moisture to drain away.

In skillet, sauté spinach in olive oil until wilted. Add cream and simmer until reduced by half, about 12 minutes. Stir in mushrooms.

On waxed paper lay one sheet of phyllo dough. Brush phyllo with butter and cover with second sheet. Brush second sheet with butter. Continue layering and brushing sheets until all five are buttered.

Place mushroom mixture on the long edge of the phyllo dough and roll toward the other edge. Fold over edges to seal. Place on an insulated baking sheet, seam side down. Place in center of cooking grate and cook over Indirect Medium heat 15 to 20 minutes or until nicely browned on top. Cut into 2-inch pieces.