Make the chutney:

1. Combine all of the chutney ingredients with the exception of the lemon and cilantro in a small saucepan. Bring to a boil. Reduce heat to low, cover, and simmer for 20 minutes. Uncover, mash the apples with a spoon (or potato masher or whisk) to help them breakdown, give everything a stir, cover and continue cooking for 10 minutes or until the mixture has thickened slightly.
2. Uncover, stir, and add the juice of half a lemon and the minced cilantro. Stir, taste, and adjust seasoning as needed: a pinch more salt or pepper flakes, a squeeze more lemon, etc. Transfer chutney to a storage vessel, let cool completely, cover and refrigerate.

Make the grilled cheese:

1. Preheat oven to 400°.  Heat an oven-proof skillet over medium-high heat. Spread a thin layer of mayonnaise over one side of each slice of bread. Add a thin layer of oil (2 teaspoons or so) to skillet. When the oil is hot, place the bread, mayonnaise side down in pan and let cook until golden, about 3 minutes. Remove pan from heat, transfer bread to work surface. 
2. Spread a layer of chutney over each non-toasted side of bread. Top with a handful of grated cheese. Place each slice of bread back into pan (cheese side up) and transfer skillet to the oven. Bake until cheese is melted, 5-7 minutes.
3. Remove skillet from oven, press the two pieces of bread together, melted cheese sides in. Halve sandwich on a diagonal and let cool briefly before eating.