Bring a large pot of water to a boil, add the frozen green beans and frozen pearl onions, cover, and cook until tender, about 3–5 minutes; drain and set aside.
Heat a large skillet over medium-high heat and add 1 tablespoon butter and the olive oil until melted and shimmering.
Add the drained green beans and pearl onions to the skillet and sauté for 1 minute, stirring to coat.
Pour in the chicken stock and cook about 1 minute, until the liquid mostly evaporates.
Reduce the heat to low, add 1 tablespoon butter (and the brown sugar if using), and gently toss until the vegetables are slightly caramelized; keep warm and move them to the side of the skillet if needed.
Increase heat back to medium-high and melt the remaining 1/2 teaspoon butter in the center of the pan; place the salmon fillets in the skillet and season as desired.
Cook the salmon 3–5 minutes on the first side until a golden crust forms, then flip and cook about 3 more minutes, until cooked through.
Plate the green beans and pearl onions and top with the cooked salmon, then serve immediately.
