Prep ingredients: Bring a medium saucepan of salted water to a boil. Halve shallot, then peel, and thinly slice lengthwise. Finely chop 1⁄4 cup of the sliced shallots (leave the rest sliced). Peel and grate 1 large garlic clove. In a medium bowl, combine beef with 2 tablespoons chopped shallot, 1 teaspoon garlic, 1⁄2 teaspoon salt, and a few grinds pepper.
Cook orzo: Add orzo to boiling water and cook until al dente, 6–8 minutes. Drain and rinse under cold water, and drain well again. Transfer orzo to a large bowl.
Make dressing: Pick dill fronds from stems and roughly chop fronds. Thinly slice roasted red peppers. Crumble half of the goat cheese. In a small bowl, combine 2 tablespoons vinegar with half of the dill, 2 tablespoons oil, 1⁄4 teaspoon salt, and a few grinds pepper.
Make burgers: Divide beef into 2 equal portions, then flatten into large, thin patties (about 5- inches wide). Using your thumb, make an indentation in the middle of each patty and divide remaining goat cheese between them. Press the meat over the cheese to make flat, round 4-inch patties. Season all over with 1⁄4 teaspoon salt and a few grinds pepper.
Cook burgers: Heat 2 teaspoons oil in a medium nonstick skillet over medium-high. Add burgers to skillet; scatter sliced shallots around the burgers. Cook burgers until lightly browned, about 3 minutes, stirring shallots occasionally. Flip burgers, continue to cook for 3 minutes more. Remove shallots from skillet once they are dark golden brown; season lightly with salt.
Finish salad & serve: To the orzo, add roasted red peppers, crumbled goat cheese, spinach, remaining chopped shallots and dill , and half of the dressing, and toss to combine; season to taste with salt and pepper. Serve burgers topped with caramelized shallots and along with the orzo salad. Drizzle burgers with remaining dressing.