Shred trout with a fork. Thinly slice 2 cloves garlic. In a 2- to 3-quart pan, combine sliced garlic, milk, and thyme sprigs; bring just to a simmer over medium heat. Remove from heat, stir in trout, cover, and let stand about 15 minutes. Remove thyme sprigs.
In a 3- to 4-quart pan, combine potatoes and about 1 quart water. Cover and bring to a boil over high heat; reduce heat and simmer until potatoes mash easily, about 20 minutes. Drain potatoes and return to pan.
Meanwhile, place four baguette slices in a single layer on a 12- by 15-inch baking sheet. Bake in a 350° regular or convection oven, turning once, until crisp and dry, about 15 minutes total. Let cool. Rub one side of each slice with remaining garlic clove. Whirl slices in a food processor into coarse crumbs, or seal in a plastic bag and crush with a rolling pin.
In a blender or food processor, whirl milk mixture until smooth. Add, with olive oil, to potatoes and beat with an electric mixer, or mash with a potato masher, until smooth. Season to taste with salt and pepper. Scrape into a shallow 1- to 2-quart baking dish.
Sprinkle top evenly with bread crumbs, then parsley. Bake in a 400° regular or convection oven until top is browned and center is hot, about 15 minutes. Serve brandade with remaining baguette slices.