Combine olive oil, vinegar,wine,salt ,pepper and garlic and put into a shallow dish.Dust off Portobellas and add to marinade.Turn in marinade periodically,filling inside of cap with marinade.Leave in marinade 30 minutes.
While mushrooms are marinating,slice peppers and onion into 1/2 inch strips and toss with olive oil ,slat and pepper.Put into a dish and set aside.
In a food processor add almonds,blend until very finely chopped.Add cheese,salt ,pepper and garlic.Process until well blended and very fine.Add basil and blend.While processing,pour in olive oil until a thick paste forms.It will usually take about 1 cup. When pesto is finished blend in mayonnaise.
Turn BBQ on to Medium heat. Place mushrooms on grill with the cup side up.Pour some marinade into each mushroom. Place other vegetables on grill. Turn occasionally until lightly brown and crisp,tender.Remove and set aside.Mushrooms will get juicy and tender.
In a 9x13 glass Baking dish place open hoagie rolls side by side. Spread pesto mayonaise on each. Slice portabellas into long strips and put into rolls.Add grilled vegetables and a small dallop of pesto mayonnaise on top of vegetables. Put 2 round pieces of provolone on top of each sandwich. Put under broiler on low just until cheese melts on top.
These are delicious served with fresh fruit and Sweet Potato Fries.