Melt the butter in a medium frying pan over low heat on the stove or side burner of the grill.
Add the shallot and garlic and cook until softened and translucent, 3–4 minutes.
Add the parsley, spinach, tarragon, and salt. Cook until the spinach has wilted.
Add the wine and bring to a simmer. Cook until nearly all the wine has evaporated.
Add the cream and bring to a simmer. Cook until the cream has reduced by about half. Stir in ¼ cup of the parmesan cheese.
Remove the mixture from the heat and top each of the shucked oysters with about 1 tablespoon of the mixture.
Combine the remaining parmesan cheese and the breadcrumbs and sprinkle it over the oysters.
Adjust the top cooking grate to the bottom setting and preheat the IR broiler on high. Working in batches of six, broil the oysters until the breadcrumbs have toasted and slightly charred, 30 seconds to 1 minute.
Serve the oysters on a platter with the lemon wedges.
