Bring a small pot with quinoa and ⅔ cup water to a boil over high heat. Reduce heat to low, 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.
Prepare the Ingredients
Peel and cut sweet potato into ¼" dice. Core Roma tomato and cut into ¼" rounds. Coarsely chop dried cherries. Zest lemon, halve, and juice. In a small bowl, combine mayonnaise with 1 tsp. lemon juice (reserve 2 tsp. for dressing) and 1 tsp. lemon zest. Split English muffins with a fork.
Roast the Sweet Potatoes
Arrange sweet potatoes on prepared baking sheet. Drizzle with 2 tsp. olive oil and season with ½ tsp. salt and ¼ tsp. pepper. Toss to coat and spread into a single layer. Roast until tender,15-18 minutes. Cool 5 minutes, and measure ½ cup sweet potato (reserve remaining for salad) into a medium mixing bowl.
Form and Sear the Patties
Using a potato masher or fork, mash sweet potato in mixing bowl until smooth. Add quinoa, breadcrumbs, and half the miso paste (reserve remaining for salad) to bowl and mix until well-combined. Season with ½ tsp. salt and ¼ tsp. pepper. Form mixture into two ½" thick patties. Heat 1 tsp. olive oil in a medium non-stick pan over medium-high heat. Add patties and sear until browned, 3-4 minutes per side.
Make Potato Salad and Toast Muffins
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 a pinch of salt and pepper. Replace foil on baking sheet and add English muffins. Bake until lightly toasted, 5-6 minutes.
Plate the Dish
Divide lemon-mayonnaise between English muffin halves and place a quinoa patty on top. Top with tomatoes, arugula, and remaining muffin halves. Serve alongside sweet potato salad.