Cover dried mushrooms with enough room temperature water so that they are fully submerged. Set aside for 35 to 45 minutes to rehydrate.
Once the mushrooms have rehydrated, trim off and dispose the woody base.
Chop into small bite-size pieces and set aside.
Meanwhile, prepare the base for the menudo.
Place the broth, chiles, 1 onion, jalapeno, and 6 cloves garlic in a pot over medium-high heat.
Bring to a boil, reduce heat to low and simmer until the chiles are softened, 5 to 7 minutes.
Remove from the heat.
Transfer everything to a blender with 1 teaspoon dried Mexican oregano and 1 teaspoon ground cumin. Pulse until smooth. Taste and adjust seasonings, adding salt and pepper, if necessary. Set aside.
Heat the vegetable oil in a large pot over medium-high heat.
Once the oil is hot, add the diced and cook, stirring often, for 1 to 2 minutes.
Add the garlic and cook for another 1 to 2 minutes.
Pour in the reserved menudo base, broth, hominy, and rehydrated snow mushrooms.
Season with the dried Mexican oregano and ground cumin, mixing to combine.
Bring to a boil, reduce the heat to medium-low, partially cover and simmer until the hominy is just tender, 20 to 30 minutes.
Add the carrots and simmer until the carrots are crisp-tender, 10 to 15 minutes.
Taste and adjust seasonings, if necessary.
Serve hot with preferred garnishes and enjoy!