Microwave the water and porcini in a covered bowl until steaming, about 1 minute. Let the mushrooms sit until softened, about 5 minutes. Drain the porcini through a fine-mesh strainer lined with a coffee filter, reserving 1/3 cup of the soaking liquid. Pulse the soaked porcini and fresh mushrooms in a food processor until they are pea-sized or smaller, about 10 pulses, scraping down the bowl as needed.

Melt butter in a large skillet over medium heat. Add the shallots and salt, cooking until the shallots are softened and translucent, about 3 to 5 minutes. Stir in the garlic and thyme until fragrant. Add the processed mushrooms and cook, stirring occasionally, until the liquid released by the mushrooms evaporates and they begin to brown, about 10 to 12 minutes.

Stir in the reserved porcini liquid and cook until it is nearly evaporated, about 1 minute. Remove the skillet from the heat and stir in the cream cheese, cream, parsley, and lemon juice until the mixture is uniform. Season with salt and pepper to taste.

Transfer the mixture to a serving bowl and smooth the top. Press plastic wrap flush to the surface of the pâté and refrigerate until firm, at least 2 hours or up to 3 days. Before serving, bring the pâté to room temperature to soften for easier spreading.