To solve this optimization problem, we need to define the decision variables and the objective function, as well as any constraints that limit the possible solutions.

Let's denote:
- \(x\) as the number of pitchers of green tea produced.
- \(y\) as the number of batches of pancakes produced.

The profit per pitcher of green tea is $2.5, and the profit per batch of pancakes is $10. Therefore, the total profit can be represented by the objective function:
\[ \text{Maximize:} \quad 2.5x + 10y \]

Now, let's consider the constraints based on the time availability for Joy and Willa:
- A pitcher of green tea takes 0.7 hours of Joy's time and 0.3 hours of Willa's time.
- A batch of pancakes takes 1.2 hours of Joy's time and 0.6 hours of Willa's time.
- Joy has 8 hours available each day.
- Willa has 5 hours available each day.

These constraints can be represented as:
\[ 0.7x + 1.2y \leq 8 \] (Joy's time constraint)
\[ 0.3x + 0.6y \leq 5 \] (Willa's time constraint)

Additionally, \(x\) and \(y\) must be non-negative since they represent the quantities of green tea pitchers and pancake batches, respectively.

Now, let's translate this problem into Gurobi code in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Tea_and_Pancakes")

# Define the decision variables
x = m.addVar(name="pitchers_of_tea", lb=0)
y = m.addVar(name="batches_of_pancakes", lb=0)

# Set the objective function
m.setObjective(2.5*x + 10*y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(0.7*x + 1.2*y <= 8, name="Joy_time_constraint")
m.addConstr(0.3*x + 0.6*y <= 5, name="Willa_time_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Pitchers of tea: {x.x}")
    print(f"Batches of pancakes: {y.x}")
    print(f"Total profit: ${2.5*x.x + 10*y.x:.2f}")
else:
    print("No optimal solution found")
```