Preheat the oven to 375°. On a large rimmed baking sheet, sprinkle the zucchini slices with salt. Arrange them in an even layer and let stand for 1 hour. In a small bowl, blend the butter with the thyme and season with salt and pepper.
Rinse the zucchini slices and pat dry; wipe off the baking sheet. Return the zucchini to the baking sheet and toss with 2 tablespoons of the olive oil. Spread the slices in an even layer and bake until golden brown on the bottom, about 30 minutes. Transfer the zucchini to a platter.
Sprinkle the sturgeon with 1 tablespoon of the vinegar and refrigerate for 10 minutes. In a large, heavy skillet, combine the remaining 2 tablespoons of vinegar with the water, bouquet garni and shallot and bring to a boil. Season the poaching liquid lightly with salt and pepper and add the sturgeon fillets. Cover tightly and simmer over low heat, turning once, until the fish is barely cooked through, about 8 minutes. Transfer the sturgeon to a large plate. Strain the poaching liquid into a bowl.
Wipe out the skillet and set it over high heat. Add the remaining 1 tablespoon of olive oil and heat until shimmering. Add the sturgeon fillets, boned side down, and cook until golden brown on the bottom, about 2 minutes. Turn and cook for 1 minute longer. Set the sturgeon on the zucchini and cover with foil to keep warm.
Add the strained poaching liquid to the skillet and boil over high heat until reduced to 1/3 cup, about 5 minutes. Stir in the capers and remove from the heat. Swirl in the thyme butter and season the sauce with salt and pepper. Transfer the sturgeon and zucchini to plates, spoon the sauce on top and serve.