Bring a small pot with quinoa and 2/3 cup water to a boil over high heat. Reduce to a simmer, cover, and cook until quinoa is tender and water has been absorbed, 12-15 minutes. Transfer quinoa to a plate and cool 5 minutes. While quinoa cooks, prepare ingredients.

Peel and cut sweet potato into 1/2" dice. Core Roma tomato and cut into 1/4" rounds. Coarsely chop dried cherries. Zest lemon, halve, and juice. In a small bowl, combine mayonnaise, 1 tsp. lemon juice (reserve 2 tsp. for dressing), and 1 tsp. lemon zest. Split English muffins with a fork.

Arrange sweet potatoes on prepared baking sheet. Toss with 2 tsp. olive oil, 1/2 tsp. salt, and 1/4 tsp. pepper and spread into a single layer. Roast until tender, 15-18 minutes. Remove from oven, transfer 1/2 cup sweet potatoes (reserve remaining for salad) to a medium mixing bowl, and cool 5 minutes.

Mash sweet potato in mixing bowl until smooth. Add quinoa, breadcrumbs, and half the miso (reserve remaining for salad) to bowl and mix until well-combined. Season with 1/2 tsp. salt and 1/4 tsp. pepper. Form mixture into two 1/2"-thick patties. Heat 1 tsp. olive oil in a large non-stick pan over medium-high heat. Add patties to hot pan and sear until browned, 3-4 minutes per side.

In another medium mixing bowl, whisk together 2 tsp. lemon juice, remaining miso paste, and 1 Tbsp. olive oil. Add remaining sweet potatoes and dried cherries to bowl and gently toss to combine. Season to taste with salt and pepper. Replace foil on baking sheet and add English muffins. Bake until lightly toasted, 5-6 minutes.

Slather lemon-mayonnaise between English muffin halves. Add a quinoa patty and top with more lemon-mayonnaise, tomato, arugula, and remaining muffin halves. Serve alongside sweet potato salad.