Melt butter in a saucepan over medium heat. Add flour and whisk to combine.
Pour milk in slowly, whisking constantly to prevent lumps from forming. Bring sauce to a simmer, stirring often.
Once the sauce begins to thicken, add cheese, and stir until melted.
Add half the jalapenos, stir to combine.
Heat a large skillet over medium heat. Melt butter in warm skillet.
Using a 2-inch circular cookie cutter, cut out the center of each piece of bread.
Place the bread slices in the skillet. Crack one egg into the center of each slice of bread.
Cook for 2 to 3 minutes per side, making sure the white is set before flipping once.
Place a piece of egg toast on your plate. Layer the toast evenly with 1/2 the avocado slices.
Spoon 2 to 3 tablespoons sauce over the avocado. Top with another slice of egg toast.
Spoon 2 to 3 tablespoons sauce over the toast and sprinkle with 1/2 the remaining jalapeno.
You could use regular sharp cheddar cheese or pepper jack for this recipe too. If you like a little more oomph to your breakfast, add a slice of tomato, some bacon, or even cooked chorizo in between the egg toast layers.