Put chopped root vegetables in a medium pot and cover with water. Bring to a boil and add 1 tsp. salt. Cook vegetables until tender, about 15 minutes.
Meanwhile, heat cream and 2 tbsp. butter in a medium saucepan over low heat. In a small bowl, combine tomato, 1 tbsp. olive oil, the vinegar, and salt and pepper to taste. Set tomato sauce aside.
Drain root vegetables. Return to pot, add heated cream, and mash until smooth. Season with salt and pepper, then set aside, covered, to keep warm.
Slice lemons in half crosswise, pop out any seeds, and brush cut sides with vegetable oil. Heat a large frying pan (not nonstick) over medium-high heat. Char lemons by setting them cut side down in pan until browned, 2 to 3 minutes. Remove from pan and set aside.
Add remaining 1 tbsp. olive oil, the shallot, and garlic to pan and cook, stirring, until sizzling, about 1 minute. Add greens, cover, and cook, stirring occasionally, until wilted and soft, about 5 minutes. Add 1 tbsp. butter and 1 tbsp. lemon juice. Cook, stirring, until butter melts. Season with salt and pepper. Keep warm, covered.
Preheat broiler with rack about 4 in. from heat. In a small frying pan over medium-high heat, melt remaining 4 tbsp. butter. Use a bit of it to butter a large baking sheet.
Open up ruby red trout like a book. If using rainbow trout, cut each into 2 fillets. Pat fish dry with paper towels, then lay skin side down on baking sheet. Drizzle 2 tsp. melted butter over each fish (1 tsp. per fillet), and sprinkle with salt and pepper. Broil until fish is cooked through and flakes easily, 5 to 10 minutes.
Meanwhile, cook remaining melted butter over medium heat, swirling pan or stirring frequently, until butter browns, about 3 minutes. Swirl in remaining 1 tbsp. lemon juice.
Divide greens and root mash among dinner plates. Arrange 1 trout (or, if using rainbow trout, 1 fillet) on each plate. Drizzle with browned butter and tomato sauce. To each plate, add a charred lemon half and some herbs or microgreens if you like.
*Ruby red trout are farm-raised, as are most rainbow trout, and fed a diet of small shrimp or crab so that their flesh is rosy, like salmon. They tend to be smaller than rainbow trout.