While the chutney is cooking, place a large skillet over medium-high heat with two turns of the pan of EVOO, about 2 tablespoons. While that's heating up, in a bowl, combine the turkey, herbs, grainy mustard and some salt and pepper. Combine everything with your hands and form it into four patties. Drop the patties into the hot pan and cook through, about 6 minutes per side.
In the last minute of cooking, lay two slices of cheese over each burger and cover the pan loosely with aluminum foil to melt the cheese.
While the burgers cook, place the English muffins split side up onto a baking sheet and toast under the broiler until golden brown (keep an eye on them so they don't burn!).
Serve the burgers up by placing a patty onto each of four muffin halves. Divide the chutney between each burger and cap them off with the remaining muffin tops.