Preheat a grill to medium-high. In a small bowl, combine cumin, garlic powder, thyme, salt, and pepper. Rub the spice blend over all of the steak pieces along with 1 tablespoon of the oil. Let rest at room temperature, about 15 minutes.
Meanwhile, in a large pot, add the remaining 2 tablespoons of oil and heat over medium heat. Add the onions and saute until tender, about 4 minutes. Stir in the mushrooms and garlic and saute until mushrooms are tender, adding extra oil if needed. Pour in the broth, jalapenos and crushed tomatoes. Bring to a boil then reduce heat and simmer while beef cooks.
On a hot grill sear steak until golden brown on both sides, about 5 minutes per side. Remove from grill to a cutting board and let rest 10 minutes. Cut beef into strips and stir into soup, along with spinach. Season soup, to taste. Transfer to serving bowls and top with a dollop of sour cream and tortilla chips.