Heat the oven to 425 F and warm a large sheet pan in it for 5 minutes.
Meanwhile, place the shallots, potatoes, carrots, olive oil, salt and black pepper in a bowl and toss to combine.
Spread out the vegetables evenly on the warm sheet pan and roast for 10 to 12 minutes.
While the vegetables cook, melt the butter in a small saucepan over medium-low heat and add in the garlic and thyme sprigs. Cook for 3 to 4 minutes until fragrant. Set aside to cool for 5 minutes and then remove the thyme sprigs and stir in the fresh lemon juice.
Pat the trout fillets dry with a paper towel and season with salt.
Drizzle the trout fillets with the garlic-thyme butter, coating all sides with your hands or a pastry brush.
Stir the vegetables and clear some space on the pan for the trout fillets.
Lower the oven temperature to 375 F and cook for 18 to 20 minutes or until the trout reaches an internal temperature of 145 F.
Remove the pan from the oven and put the roasted vegetables in a clean bowl.
In a small bowl, whisk the balsamic vinegar and honey together and toss it with the roasted vegetables. Then add the fresh parsley and dill.
Transfer the vegetables to a serving dish and top with the trout fillets. Garnish with the remaining fresh herbs and the fresh lemon slices.