Prepare the salmon glaze: In a small bowl, whisk together brown sugar, soy sauce, Dijon mustard, lemon juice, garlic, smoked paprika, salt, and pepper.
Cook the salmon: Heat olive oil in a large skillet over medium-high heat. Place salmon fillets skin side down and cook for 3–4 minutes until golden brown. Flip, then pour glaze over the fillets. Reduce heat to medium and cook another 4–5 minutes, spooning glaze over salmon until it’s caramelized and the fish flakes easily with a fork. Remove salmon and keep warm.
Prepare the garlic butter green beans: In the same skillet (wipe clean if needed), melt butter over medium heat. Add garlic and sauté for 30 seconds until fragrant. Toss in green beans, season with salt, pepper, and red pepper flakes. Cook for 4–5 minutes until tender-crisp.
Assemble and serve: Plate salmon with a side of garlic butter green beans. Spoon extra glaze over salmon and garnish with lemon wedges.