Yes, you must pay your employee (if the person is indeed an employee) to read the manual.
This is because, (1) as you note in your question, reading the manual is a state OSHA required part of the job, and (2) hourly employees must be paid at least minimum wage for all hours they work and there is no exception for time spent reading safety manuals.
Furthermore, household cooks who are employees who are paid by the hour are not exempt from the minimum wage. There is no really plausible exception to the requirement to pay a minimum hourly wage that would apply to someone who is an employee in this position. For example, this person could not plausible be classified as a salaried employee or manager in this scenario for FLSA purposes.
You could, of course, set a reasonable expectation regarding how much time should be allocated to reading the manual, so long as the expectation you set is indeed reasonable - five minutes would be unreasonably short, but 20 hours to memorize it would be unreasonably long.
Is The "Employee" An Independent Contractor?
It is conceivable that the person you describe as an employee is really an independent contractor, rather than a employee, and hence not subject to minimum wage laws.
For example, you do not have a duty to determine that a catering company that serves many customers and is its own business providing food for you on an irregular basis, or a third-party restaurant that delivers either itself or through a third-party delivery service, is paying minimum wage or complying with work safety laws like OSHA, because they are independent contractors.
But, there is nothing in your question that suggests that this is the case. (Note that if this person is an "employee" and not an "independent contractor" that it is also important that you obtain worker's compensation and pay withholding taxes for your domestic employee; often this is done via a third-party payroll service.)
The Fair Labor Standards Act of 1938 as amended, 29 U.S.C. § 201, et seq., requires you to pay every hourly employee the federal minimum wage, plus overtime if applicable, from "Portal to Portal" (i.e. from showing up to work until leaving work, because commuting time is not counted as work for purposes of the FLSA).
Hours worked is defined as follows by the FLSA at 29 U.S.C. § 203(o), contains no exception for time spent reading a safety manual:
In determining for the purposes of sections 206 and 207 of this title
the hours for which an employee is employed, there shall be excluded
any time spent in changing clothes or washing at the beginning or end
of each workday which was excluded from measured working time during
the week involved by the express terms of or by custom or practice
under a bona fide collective-bargaining agreement applicable to the
This states with respect to employees in domestic service at 209 U.S.C. § 206(f):
(1) who in any workweek is employed in domestic service in a household
shall be paid wages at a rate not less than the wage rate in effect
under subsection (b) of this section unless such employee’s
compensation for such service would not because of section 209(a)(6)
of the Social Security Act 42 U.S.C.A. § 409(a)(6)constitute wages
for the purposes of title II of such Act [42 U.S.C.A. § 401 et seq.],
(2) who in any workweek—
(A) is employed in domestic service in one or more households, and
(B) is so employed for more than 8 hours in the aggregate, shall be
paid wages for such employment in such workweek at a rate not less
than the wage rate in effect under subsection (b) of this section.
The referenced Section 206(b) is the currently effective federal minimum wage (currently $7.25 per hour). The cross-reference to the Social Security Act states that employment taxes paid by the employer (such as the employer part of FICA taxes) do not count towards paying the minimum wage.
California has a parallel state minimum wage of $10 in 2017 which is substantially similar except in amount.