Slice apples horizontally into thick round wedges and remove seeds using fork.
Remove portobello stems.
In small bowl, combine 1/8 cup olive oil, lemon juice, garlic, oregano and salt.
Using basting brush, coat portobellos on both sides with olive oil mixture. Grill stem sides down 2 minutes then flip. Add goat cheese to portobellos and grill 2-3 minutes until cheese is melted. Remove from grill.
Use remaining olive oil to coat apple rounds. Grill apple rounds 1-2 minutes per side to lightly char.
Serve portobellos with apple rounds on brioche buns with lettuce and mayonnaise.