Preheat oven to broil.
For the fig jam: Slice and discard fig stems. Cut figs into quarters. In a medium saucepot, combine figs, orange juice and zest, brown sugar, lavender and a pinch of salt over medium-high heat, and stir to combine. Cook until mixture reaches a gentle boil, about 3 minutes. Turn heat down to low and simmer for 10 minutes, stirring every few minutes. With a potato masher or fork, break figs up so the consistency is jam-like and slightly chunky. Scoop into a bowl and place in refrigerator to cool.
Place English muffins in a toaster and toast until golden brown (these will be done by the time the eggs are finished). For the eggs: In a non-stick skillet, heat butter on medium high heat. Once butter starts to sizzle, pour egg mixture into the pan. As the outside begins to cook, lift the eggs up with a spatula so that the runny center flows to the sides of the pan. When the bottom is cooked—about 30 seconds—carefully flip the eggs so that they’re now top side down.
If you’re not comfortable flipping the egg, scramble it with a spatula until soft and fluffy, about 15-20 seconds after the bottom is cooked. Immediately remove the eggs from heat.
Building the sandwiches: Spread one half of each toasted English muffin with 1 tablespoon fig jam, 1 scrambled egg portion and 1 slice of Havarti. Place sandwiches on baking tray and put under broiler until cheese is melted, about 1 minute. Remove from oven, top melted cheese with a pinch of the arugula and the other half of the English muffin. Serve with fruit.