Make the Jam. Peel the peaches, remove the pits, then chop the peaches into 1″ pieces. In a medium saucepan over medium heat, combine the peaches, light brown sugar, thyme sprigs, lemon juice, apple cider vinegar, and salt. Cook 10–15 minutes, stirring and smashing as the peaches soften, until thick and jam-like.Note: If they seem to be cooking too quickly, turn the heat down to medium-low.
Combine the Cheeses. In a mixing bowl, mix the shredded gouda and havarti cheeses together until combined. Set aside.
Build the first Sandwich Layers. Heat a large nonstick skillet over medium heat. Spread about two teaspoons of mayonnaise (or softened butter) over one side of a slice of bread. Place the bread, mayonnaise-side-down, into the hot skillet. Top with 1/4 cup of the shredded cheese blend, two slices of deli ham, and 1 tbsp. peach thyme jam.
Add more Layers. Add 1/4 cup of shredded cheese, another two slices of ham, and another tablespoon of jam. Finish with another 1/4 cup of shredded cheese.
Top with Bread and Flip. Spread mayo on another slice of bread and place it mayo-side up on the sandwich. Cook for 2–3 minutes until the bottom slice of bread is golden brown, then press down with a spatula to help the layers stick. Carefully flip the sandwich.
Finish Cooking. Cook another 2–3 minutes until the bottom bread is golden brown and crispy. (Pro tip: Cover the pan with a lid for 1–2 minutes to trap the heat and melt the cheese all the way.)
Serve. Repeat to make the second sandwich. Serve sandwiches hot and enjoy!