Kosher salt and freshly ground black pepper, to taste
2 tablespoons unsalted butter, at room temperature
4 slices whole wheat bread
1 cup shredded cheddar cheese
Heat olive oil in a large skillet over medium high heat. Add sausage and cook until browned, about 3-5 minutes, making sure to crumble the sausage as it cooks; drain excess fat, reserving 1 tablespoon in the skillet.
Add eggs to the skillet and cook, whisking, until they just begin to set. Gently whisk in half and half; season with salt and pepper, to taste. Continue cooking until thickened and no visible liquid egg remains, about 3-5 minutes; set aside.
With a butter knife, spread butter on one side of each slice of bread. Place 2 slices, buttered side down, on work surface. Divide sausage, eggs and cheese evenly on both slices. Top with remaining slices, buttered side up.
Heat the skillet over low heat. Add sandwiches and cover; cook, turning once with a spatula, until golden brown and the cheese has just melted, about 2-3 minutes per side.