Heat the butter and oil in a large cast iron skillet over medium-high heat until the butter foams. Add half the corn in a single layer and leave it alone for 3 to 4 minutes, until the bottom kernels are blistered and browned in spots. Stir and cook 1 to 2 minutes more. Move to a bowl and repeat with the remaining corn.
Return all the corn to the pan, add the jalapeño and garlic, and cook for 1 minute, just until fragrant. Spread the corn on a plate and let it cool for about 10 minutes.
In a large bowl, beat the softened cream cheese until smooth, then stir in the mayonnaise, sour cream, lime juice, lime zest, chili powder, cumin and salt.
Fold in the cooled corn, 1/2 cup of the cotija, most of the cilantro and the green onions. Taste and adjust with more lime, salt or chili powder.
To serve cold: spoon into a serving bowl, cover and chill for at least 30 minutes. Top with the remaining cotija, cilantro and a pinch of chili powder just before serving.
To serve hot: heat the oven to 375°F. Spread the dip in a 1-quart baking dish (or leave it in the oven-safe skillet), scatter the Monterey Jack on top and bake for 15 to 20 minutes, until bubbling at the edges. Finish with the remaining cotija, cilantro and chili powder.