Combine tomatoes, cucumber, shallot, mint, sugar, salt, pepper, and vinegar in a small bowl. Whisk in olive oil and set aside at room temperature at least one hour before serving.
Pat fish dry with paper towels and season with salt and pepper. Heat 2 tablespoons vegetable oil in a 12-inch or larger saute pan over medium-high heat. Add fish to pan (in batches if necessary) and cook for 3 to 4 minutes per side until nicely browned and cooked through. Just before removing from pan, add butter and cook for an additional 30 seconds until butter almost browns; the fish should flake easily when pressed. Repeat with remaining fish, adding more vegetable oil and butter to saute pan as needed. Garnish with sea salt.
Serve immediately with spoonfuls of vinaigrette over the top and drizzled with more olive oil, as desired.