Preheat the oven to 400°F (204°C) and line a sheet pan with parchment paper.
Trim the ends of the oyster mushroom stems and tear the mushrooms into small shreds by hand.
In a small bowl whisk together vegetable oil, tikka masala, ground cumin, turmeric, cayenne, and salt to make the marinade.
Place the shredded mushrooms in a large mixing bowl and pour the marinade over them; toss gently to coat every piece evenly.
Arrange the marinated mushrooms in a single layer on the prepared sheet pan.
Roast in the preheated oven for about 20–30 minutes, flipping once halfway through, until edges are browned and slightly crisp—cooking time varies with piece size.
While still hot, combine the roasted mushrooms with the prepared vegan tikka masala sauce so the sauce warms and coats the mushrooms.
Slice each sandwich bun in half and layer with lettuce leaves.
Spoon a generous portion of the sauced mushroom strips onto each bun half and assemble the sandwiches.