Heat olive oil in a medium nonstick skillet over medium heat; add apples, onion and jalapeños; sauté for 8 minutes. Add garlic, all spice and chili powder; sauté 1 minute more. Add apple cider vinegar and brown sugar, stirring until all liquid is absorbed, about 3 to 4 minutes.
Preheat oven to 225° F.
Lay 4 tortillas onto work surface. Top each with all of the apple chutney and blue cheese; cover with remaining tortillas. Brush tortilla tops with olive oil. Working in batches, place 1 quesadilla in large nonstick skillet over medium heat, oiled side down. Cook until bottom is golden, about 3 minutes. Brush top of quesadilla with olive oil and flip to cook other side until golden, about 3 minutes more. Place cooked quesadillas on large cookie sheet and hold in oven while cooking remaining quesadillas. Cut quesadillas into wedges, place on a decorative platter and add a dollop of sour cream to each.