Adjust rack to top position and preheat oven to 425 degrees. Bring 2 TBSP butter (3 TBSP for 4 servings) to room temperature. Wash and dry all produce. Cut potatoes into 1/2-inchthick wedges. Pick tarragon leaves from stems; discard stems. Mince leaves until you have 1 TBSP (2 TBSP for 4). Quarter and core apple. Grate 1 quarter (2 quarters for 4) on the largest holes of a box grater; thinly slice remaining apple.

Toss potatoes on a baking sheet with a large drizzle of olive oil, half the Herbes de Provence (you’ll use the rest later), and a generous pinch of salt and pepper. Roast until golden brown and tender, 20-25 minutes.

Meanwhile, in a large bowl, whisk together 1 TBSP olive oil, half the vinegar, half the mustard, 1/2 tsp minced tarragon, 1/2 tsp sugar, salt, and pepper. (For 4 servings, whisk together 2 TBSP olive oil, all the vinegar, half the mustard, 1 tsp minced tarragon, 1 tsp sugar, salt, and pepper.)

In a small bowl, combine mayo, remaining mustard, and 11/2 tsp minced tarragon (3 tsp for 4 servings). Season with salt and pepper.

In a second large bowl, combine beef, grated apple, remaining minced tarragon, remaining Herbes de Provence, 3/4 tsp salt (11/2 tsp for 4 servings), and pepper. Form into two equal-sized patties (four patties for 4), each slightly wider than a burger bun. Heat a drizzle of oil in a large pan over medium-high heat. Add patties; cook to desired doneness, 3-5 minutes per side. In the last 2 minutes of cooking, top patties with cheese; cover pan to melt.

Halve buns and spread with softened butter; toast until golden brown. To bowl with vinaigrette, add arugula (reserving a handful for topping burgers), sliced apple, almonds, salt, and pepper; toss to coat. Spread half the aioli onto toasted buns. Fill with patties and reserved arugula. Serve with potato wedges, salad, and remaining aioli on the side.