For the pickled jalapenos: Pack the jalapenos, garlic, onion, and thyme into a 2-quart jar. Combine the vinegar, water and 1/2 cup salt in a medium saucepan; bring to a boil. Pour over the jalapenos and refrigerate 3 to 4 days.

For the English muffins: In the bowl of an electric mixer, combine the yeast and water; set aside until foamy, about 5 minutes. Stir in the flour, butter, 1 teaspoon salt, and sugar. Using the dough hook attachment on the electric mixer, mix the dough on medium speed until smooth and firm, about 10 minutes. Shape the dough into a ball; place into a greased bowl. Let rise, covered, until doubled in bulk, 1 1/2 to 2 hours. 

Divide the dough into 12 equal portions. Knead the dough into balls; dust the work surface with cornmeal. Press each dough portion into a 3 1/2-inch round cutter over cornmeal, then transfer to a baking sheet. Cover and let rise 30 minutes. 

Preheat the oven to 400 degrees F. Heat a large cast-iron skillet over medium-high heat; working in batches, add English muffins. Cook 5 minutes; transfer to a separate baking sheet. Transfer the muffins to the oven to finish cooking, 2 to 5 minutes. 

In a medium, microwave-safe bowl, whisk together the eggs and oil; season with salt. Microwave on high, 50 to 60 seconds. Transfer the eggs to a large nonstick skillet over medium heat; cook until desired doneness. 

Split 4 English muffins; spread each with 1 tablespoon mayonnaise. Top muffins with egg, pickled jalapenos, sausage and cheese; serve warm.