Melt the butter in a 12-inch skillet over medium heat. Add the onion and thyme and cook, stirring occasionally, until the onion is dark brown, 15 to 20 minutes—it should be darker than golden, but not black. Stir in the cremini and shiitake, 1/2 teaspoons salt, and 1/2 teaspoons pepper; cook, stirring occasionally, until the mushrooms are soft and golden and the liquid they release has evaporated, 8 to 10 minutes. Add the sherry and cook, scraping up the browned bits from the pan, until almost evaporated, about 1 minute. Let the mixture cool to room temperature.
Transfer half of the mushroom mixture to a food processor. Add the walnuts, garlic, lemon juice, zest, 3/4 teaspoons salt, and 1/2 teaspoons pepper. Process until smooth, stopping occasionally to scrape down the sides of the bowl. Add the remaining mushroom mixture and the eggs. Pulse until coarsely chopped. Season to taste with more salt, pepper, and lemon juice. Serve in a crock or bowl or mounded on a platter, drizzled with olive oil and sprinkled with flaky sea salt and scallion greens or chives.