Rinse fish and pat dry; coat pieces with about 2 teaspoons olive oil. Sprinkle lightly all over with salt and pepper. Set a 10- to 12-inch nonstick frying pan with ovenproof handle over high heat; when hot, set fish in pan and cook until lightly browned on the bottom, 2 to 3 minutes.
Transfer pan with fish to a 350° regular or convection oven and bake until fish is barely opaque but still moist-looking in center of thickest part (cut to test), 7 to 9 minutes.
Meanwhile, rinse and drain mushrooms; trim off and discard stems for shiitakes (tough stem ends only for portabellas). Cut mushrooms into 1/2-inch chunks.
In an 8- to 10-inch nonstick frying pan over medium-high heat, stir mushrooms and garlic in 1 teaspoon oil until mushrooms are limp, about 5 minutes. Add broth, tomato sauce, and soybeans; simmer, uncovered, stirring occasionally, until soybeans are tender to bite, about 5 minutes. Stir in 1 tablespoon chives. Add salt and pepper to taste.
Mound a fourth of the soybean mixture in the center of each of four dinner plates. Top each mound with a piece of fish, pan-browned side up, and spoon Kabocha coulis equally around soybean mixture. Gently shake each plate to level coulis. Garnish fish with remaining tablespoon chives and scatter chopped basil over coulis if desired. Add more salt and pepper to taste.