Step 1
Place 2 Tbsp. black peppercorns on a cutting board or seal in a resealable plastic bag and coarsely crush with a large stainless-steel skillet or heavy pot. (Or, coarsely chop with a knife. They should be a lot coarser than what you would typically season with.)
Step 2
Toss 2 lb. mixed mushrooms, halved, torn into 1″ pieces if large, ¼ cup extra-virgin olive oil, 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt, and 2 tsp. crushed peppercorns in a large bowl to combine.
Step 3
Heat 2 Tbsp. extra-virgin olive oil in a medium cast-iron skillet over medium-high. Add half of mushrooms; top with a steak weight, small cast-iron skillet, or small heavy saucepan or pot. Cook until mushrooms release their juices and they are simmering, 5–7 minutes. Remove weight and cook until liquid is evaporated, about 3 minutes. Turn mushrooms over the best you can (you might have some broken pieces) and cook, stirring occasionally, until golden brown, about 4 minutes. Using a slotted spoon, transfer to a plate. Repeat process with remaining mushrooms.
Step 4
Return all mushrooms to pan and reserve plate. Add 5 sprigs thyme, 4 garlic cloves, smashed, and 5 Tbsp. unsalted butter to mushrooms and cook, stirring often, until mushrooms are deeply browned and garlic and thyme are fragrant, about 4 minutes. Transfer mushrooms to reserved plate with slotted spoon.
Step 5
Add 1 large shallot, finely chopped, and remaining crushed peppercorns to fat in pan; cook, stirring constantly, until shallot is softened, about 3 minutes. Remove pan from heat and carefully pour in 1½ cups cognac, dry sherry, brandy, or low-sodium vegetable broth. Return to heat and cook, stirring occasionally, until reduced by half, about 4 minutes. Add ½ cup heavy cream, 1 Tbsp. soy sauce, and ½ tsp. balsamic vinegar; cook until sauce is thick enough to lightly coat a spoon, about 3 minutes. Add remaining 3 Tbsp. unsalted butter; stir until melted and emulsified.
Step 6
Remove pan from heat; season sauce with more salt and freshly ground pepper if needed. (It should taste strongly of pepper.) Return mushrooms to pan; top with thinly sliced chives if desired.
Step 7
Place mixed greens in a large bowl. Lightly drizzle with oil and fresh lemon juice, season with salt and pepper, and toss to coat. Serve mushrooms with salad and country-style bread alongside.