Use a jaccard steak tenderizing tool, or tenderize the steak yourself with a fork. Lay the meat out in front of you, pushing it flat with your hands or a meat mallet, and pull it gently apart with tines of the fork until the meat is slightly pulled apart and tender. Season with salt and pepper and dredge in flour. In a large oven-safe skillet over medium-high heat, add olive oil and brown steaks, in batches if necessary, on both sides, about 2 minutes per side. Set aside, keep warm.
Meanwhile, in a saucepan, reduce stock by half.
Season mushrooms with salt and pepper, to taste. Saute quartered mushrooms in the same pan as the meat, in the leftover olive oil, until very brown, about 5 minutes. Add butter, minced onion, carrot, thyme, and garlic. Cook until vegetables are caramelized, about 5 minutes, add balsamic vinegar and reduce for 1 to 2 minutes. Add broth, slide meat back into sauce, cover and bake until tender, about 1 hour.
Stack steaks with pea whipped potatoes and drizzle with balsamic/mushroom sauce.
Pea Whipped Potatoes:
2 pounds russet potatoes, peeled and cut into large chunks
1 1/2 pounds English peas, or 2 pounds frozen peas
2 cups heavy cream
1/2 cup (2 sticks) unsalted butter, at room temperature
1 tablespoon minced garlic (about 3 to 4 cloves)
Salt and freshly ground black pepper
Grated Parmesan, optional
Put the potatoes in a large saucepan with cold water to cover by 3 inches. Salt the water, cover and bring to a boil over high heat. Reduce heat and simmer until tender, about 10 minutes. While the potatoes are cooking, cook the peas in a separate saucepan of salted water. Simmer until just tender, about 3 to 5 minutes. Remove with a sieve and cool on a baking sheet. When the potatoes are done, drain well, scatter on a baking sheet, and let cool for several minutes. In a small saucepan, heat 1/2 cup of the cream until it reaches a boil then simmer for a few minutes. Add the cream to a blender with the cooked peas and puree until smooth. Season with salt and pepper.
Heat 1 tablespoon of the butter in a large saucepan over medium heat until the butter stops foaming and turns a light brown. Add the garlic and cook quickly until pale gold. Add remaining cream, season, to taste, with salt and pepper and bring to a boil.
Press the cooled potatoes through a ricer or food mill into the butter and garlic mixture, tossing to coat.
Reduce the heat to low. Add the remaining butter by tablespoonfuls, stirring vigorously after each addition. In this way, the butter does not melt into a golden puddle. Instead, it forms an emulsion with the potatoes, creating a very smooth puree.
Fold the pea puree into the mashed potatoes and adjust seasoning, as necessary. Instead of salt, the puree can be seasoned with a little grated Parmesan, if desired.