Heat the oven to 375°F (190°C). Line a rimmed sheet pan with parchment. Whisk all glaze ingredients together until smooth and set aside.
Heat olive oil in a wide skillet over medium-high. Cook the mushrooms for 6–8 minutes until their liquid evaporates. Add the onion, reduce heat to medium, and cook for 7–9 minutes. Stir in the garlic and tomato paste for 60 seconds. Transfer to a plate and cool for 15 minutes.
Combine the torn bread and milk in a large bowl. Let stand 5 minutes, then mash with a fork until no dry center remains. Add the eggs, salt, pepper, thyme, paprika, parsley, and cooled mushroom mixture, and whisk gently to combine.
Add the beef in loose pieces. Fold with damp hands for 45–60 seconds, stopping as soon as the mixture looks even.
Shape on the prepared pan into a loaf about 25 cm long, 14 cm wide, and 6 cm high. Bake for 35 minutes. Spoon away any large pool of rendered fat, then spread two-thirds of the glaze over the top and sides. Bake for 15 minutes.
Apply the remaining glaze. Bake 5–15 minutes more until the center reaches 160°F (71°C). Rest for 15 minutes before slicing into 8 portions.