Preheat the oven to 375 degrees F.
Arrange the bacon on a sheet tray, sprinkle with some of the thyme and bake until crispy, about 12 to 15 minutes.
Slice the apples horizontally to create 5 to 6 cross sections of the apple. On a separate sheet tray lined with a silpat or parchment paper lay the apples out in a single layer and drizzle with a little olive oil. Sprinkle a little sugar and thyme on each slice then bake in the oven until the apples begin to dehydrate and sugar has melted and caramelized, about 8 to 10 minutes
To serve, lay the apple and bacon slices, alternating, in a long overlapping line on a serving platter. Or, break the bacon pieces in half and reassemble the apple slices back together, placing crispy bacon pieces in between each slice. Garnish with arugula and serve.