Thaw the spinach in a microwave, then wring dry in a clean towel.
Preheat a griddle or cast-iron skillet over medium-high heat. Season the steaks with salt and pepper and coat with some EVOO. Cook, turning once, for 10 minutes for medium rare. Transfer to a cutting board and let rest for 10 minutes; thinly slice.
Meanwhile, in a small saucepan, heat 2 tbsp. EVOO, 2 turns of the pan, over medium-high heat. Stir in the shallot and garlic for 2 minutes; season with a little salt and lots of pepper. Add the ketchup, beef stock, worcestershire, vinegar, brown sugar and porcinis. Bring to a boil, then lower the heat and simmer for 20 minutes. Puree the mixture in a blender.
Preheat the oven to 500 degrees . Halve the tomatoes and toss with the remaining 2 tbsp. EVOO and the thyme. Place the tomatoes cut side down on a parchment-lined baking sheet and roast until slumped and the skins are charred, 20 to 25 minutes.
Add the cream to a medium saucepan. Add the onion and bring to a low bubble. Stir in the cheese to melt. Add the spinach; season with the nutmeg, salt and pepper. Lower the heat to low until ready to serve.
Top the steaks with the sauce and serve with the tomatoes and spinach. (For an alternative presentation, fan out the steaks.)