Melt 1 tablespoon unsalted butter in a large pot over medium-low heat. Add 1 sweet onion (diced), 1 cup diced carrots, 4 garlic cloves (minced), a pinch of kosher salt and pepper, and ½ teaspoon dried dill. Cook, stirring often, about 5 minutes, until the vegetables soften.
Add 4 cups chicken stock and 1 pound baby Yukon gold potatoes (cut in half). Increase heat to bring to a boil, then reduce to a simmer and cook until the potatoes are tender, about 10–15 minutes.
While the potatoes cook, make the dumpling batter: In a bowl whisk together 1 cup all-purpose flour, 2 teaspoons baking powder, a pinch of kosher salt and pepper, ½ teaspoon dried dill, ½ teaspoon dried chives, and ½ teaspoon garlic powder. In a smaller bowl whisk 1 large egg (lightly beaten) with ½ cup milk, then pour the milk mixture into the dry ingredients and stir until just combined. Set the batter aside.
Make a slurry: In a jar or small bowl combine 1 tablespoon all-purpose flour with the remaining 1 cup chicken stock and whisk or shake until smooth.
When the potatoes are tender, slowly whisk the slurry into the pot. Bring the soup back to a gentle simmer.
Stir in 1 cup frozen corn and ⅓ cup heavy cream. Gently add 1 pound salmon (cut into chunks) and stir to combine.
If you are not adding dumplings: simmer the salmon gently until cooked through, about 5 minutes. Taste and season with kosher salt and pepper as needed. Top with sliced green onions and serve.
If you are adding dumplings: using a spoon, drop the dumpling batter into the simmering soup in rounded 2-tablespoon scoops, spacing them apart. Cover the pot and simmer for 8 minutes.
After 8 minutes, uncover and gently flip each dumpling with a spoon. Continue simmering, uncovered, until the dumplings are cooked through and the salmon is done, about 5 more minutes. Taste and adjust seasoning with kosher salt and pepper.
Ladle the chowder into bowls, top with sliced green onions, and serve.