Slice each of the 4 pieces of ciabatta bread horizontally in half.
Heat a skillet over medium-low heat; add oil. Cook and stir onion in the hot oil over low heat until onion begins to soften and lightly brown, 10 to 15 minutes. Add mushrooms; cook and stir until tender and browned, 5 to 10 minutes more. Add pea shoots and cook until wilted, 2 to 3 minutes. Remove skillet from heat.
Open the ciabatta pieces, creating 4 sandwiches. Evenly distribute the pea shoot mixture among the bottom pieces of each sandwich; add 1 slice Muenster cheese and 3 slices deli ham. Replace the top pieces of ciabatta to make a sandwich.
