Heat 2 tablespoons olive oil in a skillet over medium-high heat. Add finely minced mushrooms and a pinch of salt. Cook and stir until mushrooms completely dry out and turn golden brown, 7 or 8 minutes, reducing heat slightly toward the end if needed.
Break bread into large chunks and place in food processor. Pulse on and off until processed into fine crumbs. Add crumbs to mushroom mixture. Cook over medium heat until breadcrumbs are completely dried out, dark brown and toasted. Drizzle in more oil if needed. Stir in ground almonds. Continue cooking on medium until pieces of almond turn brown, 3 to 4 minutes. Remove from heat and allow "dirt" mixture to cool to room temperature.
Scoop spoonfuls of burrata into serving bowls. Arrange tomato slices on top. Drizzle with white wine vinegar and a generous drizzle of olive oil. Season with kosher salt, black pepper, and basil. Sprinkle with "dirt" mixture.