How to Make It
Preheat oven to 400°F. Add potatoes, 2 tablespoons salt, and water to cover to a medium saucepan; bring to a boil over high. Reduce heat to medium-high, and simmer, uncovered, just until potatoes are tender when pierced with a knife, about 10 minutes. Using a slotted spoon, transfer potatoes to a medium bowl, reserving water in pan. Pour 1/4 cup wine over potatoes. Immediately remove potatoes (do not let sit in wine), and cut into 1/2-inch-thick coins. Set potatoes aside; reserve wine. Add carrots to hot water in pan, and return to a boil over high. Cook until crisp-tender, about 6 minutes; drain carrots, and set aside.
Melt 3 tablespoons butter in a large skillet over medium-low. Add leeks; stir to coat. Add ginger and 3/4 teaspoon salt; cook, stirring occasionally, just until crisp-tender but not browned, 8 to 10 minutes. Remove from heat; cover to keep warm.
Place shallots and remaining 3/4 cup wine plus reserved wine from potatoes in a small saucepan, and bring to a boil over medium-high. Boil, undisturbed, until wine is reduced to about 2 tablespoons, 8 to 10 minutes. Remove from heat, and whisk in crème fraîche until blended. Return to heat over low. Gradually whisk in 1/2 cup butter, 1 piece at a time, whisking until almost emulsified before adding next piece. Whisk in lemon juice and 1/4 teaspoon salt. Remove from heat; cover and keep warm.
Heat canola oil in a large cast-iron skillet over medium-high just until smoking. Sprinkle hake fillets with remaining 1 1/2 teaspoons salt and flour. Add fillets to skillet, skin side down; cook, gently pressing fillets with a spatula, until skin begins to brown, about 3 minutes. Transfer to preheated oven, and roast 3 minutes. Return to stovetop over high. Carefully turn fish over, and add remaining 1/4 cup butter. Tilt skillet toward you so butter pools, and cook, basting fish often, until fish is just cooked through, about 2 minutes. Remove from heat.
Divide potatoes, carrots, and fish among 4 plates. Spoon leek mixture and white wine sauce over and around fish and vegetables. Top with kumquats and chives.