Preheat the oven to 375 degrees F (190 degrees C). Very generously grease a loaf pan.

Heat olive oil in a large skillet over high heat. Add mushrooms and 1 teaspoon kosher salt and sauté until nicely browned and juices have evaporated, about 10 minutes. Stir in onion and cook until soft, another 5 to 7 minutes.

Add celery, tomato paste, and beef base; cook and stir until tomato paste begins to toast on the bottom of the pan, about 3 minutes. Stir in water and turn off the heat; let sit for about 3 minutes.

Transfer mushroom mixture into a food processor. Add cooked lentils and pulse until finely chopped but not pureed.

Transfer to a mixing bowl and add eggs, soy sauce, Worcestershire sauce, onion powder, garlic powder, pepper, remaining 1 teaspoon salt, oregano, thyme, cayenne, ketchup, bread crumbs, oats, and Parmigiano-Reggiano cheese. Mix until thoroughly combined.

Transfer mixture into the prepared loaf pan, then wrap with foil and place on a sheet pan.

Bake in the center of the preheated oven until firm to the touch, about 1 hour.

Remove the loaf from the oven and place on a wire rack. Remove the foil and place it on the baking sheet. Grease the foil with 2 teaspoons olive oil. Let the loaf sit for 10 to 15 minutes, and increase the oven temperature to 450 degrees F (230 degrees C).

While the loaf is resting, make the glaze: Whisk ketchup, Dijon mustard, and brown sugar together in a small bowl.

Invert the loaf onto the foil and spread glaze all over the surface, mostly on the top but some on the sides.

Return to the oven and bake until the glaze is cooked on, 15 to 20 minutes.

Carefully transfer the loaf onto a cutting board. Cut into 6 slices and serve.