Heat the oven to 375°F. Line a rimmed baking sheet with parchment paper.
Warm the olive oil in a skillet over medium heat. Add the onion and mushrooms, then cook for 6 to 8 minutes until softened and their liquid has mostly evaporated. Stir in the spinach and one minced garlic clove. Cook until the spinach wilts, then transfer the mixture to a plate to cool.
In a mixing bowl, combine the ground turkey, egg, breadcrumbs, 2 tablespoons milk, thyme, ½ teaspoon salt, and ¼ teaspoon black pepper. Mix gently just until combined.
Place the turkey mixture on the prepared baking sheet and shape it into a rectangle about ½ inch thick. Spread the cooled spinach and mushroom mixture down the center, leaving a border on all sides. Fold the long sides over the filling and pinch the seam and ends closed. Turn the loaf seam-side down.
Bake for 40 to 50 minutes, until the center reaches 165°F. Let the meatloaf rest for 10 minutes before slicing so the filling stays in place.
While the meatloaf bakes, put the potatoes in a saucepan and cover them with water. Bring to a boil, then simmer for 15 to 20 minutes until fork-tender. Drain well.
Return the potatoes to the saucepan. Add the butter, remaining minced garlic, remaining ¼ teaspoon salt, remaining ¼ teaspoon black pepper, and ¼ cup milk. Mash until smooth, adding a little more milk only if needed for your preferred texture.
Slice the rested meatloaf and serve with the garlic mashed potatoes.